Calculus Examples

$f (x) = x \sqrt{9 - x}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x}$ as ${(9 - x)}^{\frac{1}{2}}$ .

$f' (x) = \frac{d}{d x} (x {(9 - x)}^{\frac{1}{2}})$

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(9 - x)}^{\frac{1}{2}}$ .

$f' (x) = x \frac{d}{d x} ({(9 - x)}^{\frac{1}{2}}) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 - x$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $9 - x$ .

$f' (x) = x (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$f' (x) = x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.3.3

Replace all occurrences of $u$ with $9 - x$ .

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.5

Combine $- 1$ and $\frac{2}{2}$ .

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.6

Combine the numerators over the common denominator.

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $- 1$ by $2$ .

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.7.2

Subtract $2$ from $1$ .

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{\frac{- 1}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.8

Combine fractions.

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Step 1.8.1

Move the negative in front of the fraction.

$f' (x) = x (\frac{1}{2} \cdot {(9 - x)}^{- \frac{1}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.8.2

Combine $\frac{1}{2}$ and ${(9 - x)}^{- \frac{1}{2}}$ .

$f' (x) = x (\frac{{(9 - x)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.8.3

Move ${(9 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.8.4

Combine $\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ and $x$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.9

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (9) + \frac{d}{d x} (- x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.10

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.11

Add $0$ and $\frac{d}{d x} [- x]$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.12

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.13

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (- 1 \cdot 1) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.14

Combine fractions.

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Step 1.14.1

Multiply $- 1$ by $1$ .

$f' (x) = \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot - 1 + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.14.2

Combine $\frac{x}{2 {(9 - x)}^{\frac{1}{2}}}$ and $- 1$ .

$f' (x) = \frac{x \cdot - 1}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.14.3

Simplify the expression.

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Step 1.14.3.1

Move $- 1$ to the left of $x$ .

$f' (x) = \frac{- 1 \cdot x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.14.3.2

Rewrite $- 1 x$ as $- x$ .

$f' (x) = \frac{- x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.14.3.3

Move the negative in front of the fraction.

$f' (x) = - \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)$

Step 1.15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = - \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \cdot 1$

Step 1.16

Multiply ${(9 - x)}^{\frac{1}{2}}$ by $1$ .

$f' (x) = - \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}}$

Step 1.17

To write ${(9 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f' (x) = - \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \cdot \frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.18

Combine ${(9 - x)}^{\frac{1}{2}}$ and $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f' (x) = - \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{{(9 - x)}^{\frac{1}{2}} (2 {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.19

Combine the numerators over the common denominator.

$f' (x) = \frac{- x + {(9 - x)}^{\frac{1}{2}} (2 {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.20

Multiply ${(9 - x)}^{\frac{1}{2}}$ by ${(9 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.20.1

Move ${(9 - x)}^{\frac{1}{2}}$ .

$f' (x) = \frac{- x + {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{\frac{1}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.20.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{- x + {(9 - x)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.20.3

Combine the numerators over the common denominator.

$f' (x) = \frac{- x + {(9 - x)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.20.4

Add $1$ and $1$ .

$f' (x) = \frac{- x + {(9 - x)}^{\frac{2}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.20.5

Divide $2$ by $2$ .

$f' (x) = \frac{- x + (9 - x) \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.21

Simplify ${(9 - x)}^{1} \cdot 2$ .

$f' (x) = \frac{- x + (9 - x) \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.22

Move $2$ to the left of $9 - x$ .

$f' (x) = \frac{- x + 2 \cdot (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23

Simplify.

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Step 1.23.1

Apply the distributive property.

$f' (x) = \frac{- x + 2 \cdot 9 + 2 (- x)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.2

Simplify the numerator.

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Step 1.23.2.1

Simplify each term.

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Step 1.23.2.1.1

Multiply $2$ by $9$ .

$f' (x) = \frac{- x + 18 + 2 (- x)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.2.1.2

Multiply $- 1$ by $2$ .

$f' (x) = \frac{- x + 18 - 2 x}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.2.2

Subtract $2 x$ from $- x$ .

$f' (x) = \frac{- 3 x + 18}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.3

Factor $3$ out of $- 3 x + 18$ .

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Step 1.23.3.1

Factor $3$ out of $- 3 x$ .

$f' (x) = \frac{3 (- x) + 18}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.3.2

Factor $3$ out of $18$ .

$f' (x) = \frac{3 (- x) + 3 (6)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.3.3

Factor $3$ out of $3 (- x) + 3 (6)$ .

$f' (x) = \frac{3 (- x + 6)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.4

Factor $- 1$ out of $- x$ .

$f' (x) = \frac{3 (- (x) + 6)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.5

Rewrite $6$ as $- 1 (- 6)$ .

$f' (x) = \frac{3 (- (x) - 1 \cdot - 6)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.6

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$f' (x) = \frac{3 (- (x - 6))}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.7

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$f' (x) = \frac{3 (- 1 (x - 6))}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 1.23.8

Move the negative in front of the fraction.

$f' (x) = - \frac{3 (x - 6)}{2 {(9 - x)}^{\frac{1}{2}}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3 (x - 6)}{2 {(9 - x)}^{\frac{1}{2}}}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [\frac{x - 6}{{(9 - x)}^{\frac{1}{2}}}]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{d}{d x} \frac{x - 6}{{(9 - x)}^{\frac{1}{2}}}$

Step 2.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 6$ and $g (x) = {(9 - x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{{({(9 - x)}^{\frac{1}{2}})}^{2}}$

Step 2.3

Multiply the exponents in ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{{(9 - x)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{{(9 - x)}^{\frac{1}{2} \cdot 2}}$

Step 2.3.2.2

Rewrite the expression.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.4

Simplify.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x - 6) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} (\frac{d}{d x} (x) + \frac{d}{d x} (- 6)) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} (1 + \frac{d}{d x} (- 6)) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.5.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} (1 + 0) - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} \cdot 1 - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.5.4.2

Multiply ${(9 - x)}^{\frac{1}{2}}$ by $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}}{9 - x}$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 - x$ .

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Step 2.6.1

To apply the Chain Rule, set $u_{1}$ as $9 - x$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.6.3

Replace all occurrences of $u_{1}$ with $9 - x$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.8

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.9

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.10.2

Subtract $2$ from $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{\frac{- 1}{2}} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.11

Combine fractions.

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Step 2.11.1

Move the negative in front of the fraction.

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2} \cdot {(9 - x)}^{- \frac{1}{2}} \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.11.2

Combine $\frac{1}{2}$ and ${(9 - x)}^{- \frac{1}{2}}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{{(9 - x)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.11.3

Move ${(9 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x))}{9 - x}$

Step 2.12

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (9) + \frac{d}{d x} (- x)))}{9 - x}$

Step 2.13

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x)))}{9 - x}$

Step 2.14

Add $0$ and $\frac{d}{d x} [- x]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x))}{9 - x}$

Step 2.15

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} - (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x))}{9 - x}$

Step 2.16

Multiply.

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Step 2.16.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} + 1 (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x)))}{9 - x}$

Step 2.16.2

Multiply $x - 6$ by $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (x)))}{9 - x}$

Step 2.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}) \cdot 1}{9 - x}$

Step 2.18

Combine fractions.

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Step 2.18.1

Multiply $x - 6$ by $1$ .

$f'' (x) = - \frac{3}{2} \cdot \frac{{(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}})}{9 - x}$

Step 2.18.2

Multiply $\frac{{(9 - x)}^{\frac{1}{2}} + (x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}}}{9 - x}$ by $\frac{3}{2}$ .

$f'' (x) = - \frac{({(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}})) \cdot 3}{(9 - x) \cdot 2}$

Step 2.18.3

Reorder.

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Step 2.18.3.1

Move $3$ to the left of ${(9 - x)}^{\frac{1}{2}} + (x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 \cdot ({(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}))}{(9 - x) \cdot 2}$

Step 2.18.3.2

Move $2$ to the left of $9 - x$ .

$f'' (x) = - \frac{3 ({(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}))}{2 \cdot (9 - x)}$

Step 2.19

Simplify.

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Step 2.19.1

Apply the distributive property.

$f'' (x) = - \frac{3 {(9 - x)}^{\frac{1}{2}} + 3 ((x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}))}{2 (9 - x)}$

Step 2.19.2

Apply the distributive property.

$f'' (x) = - \frac{3 {(9 - x)}^{\frac{1}{2}} + 3 ((x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}))}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3

Simplify the numerator.

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Step 2.19.3.1

Factor $3$ out of $3 {(9 - x)}^{\frac{1}{2}} + 3 (x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ .

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Step 2.19.3.1.1

Factor $3$ out of $3 {(9 - x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 ({(9 - x)}^{\frac{1}{2}}) + 3 (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.1.2

Factor $3$ out of $3 (x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{3 ({(9 - x)}^{\frac{1}{2}}) + 3 ((x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}))}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.1.3

Factor $3$ out of $3 ({(9 - x)}^{\frac{1}{2}}) + 3 ((x - 6) \frac{1}{2 {(9 - x)}^{\frac{1}{2}}})$ .

$f'' (x) = - \frac{3 ({(9 - x)}^{\frac{1}{2}} + (x - 6) (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}))}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.2

Let $u_{2} = {(9 - x)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(9 - x)}^{\frac{1}{2}}$ .

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Step 2.19.3.2.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{3 (\frac{2 u_{2} u_{2} + x - 6}{2 u_{2}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.2.2

Multiply $u_{2}$ by $u_{2}$ by adding the exponents.

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Step 2.19.3.2.2.1

Move $u_{2}$ .

$f'' (x) = - \frac{3 (\frac{2 (u_{2} u_{2}) + x - 6}{2 u_{2}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.2.2.2

Multiply $u_{2}$ by $u_{2}$ .

$f'' (x) = - \frac{3 (\frac{2 {u_{2}}^{2} + x - 6}{2 u_{2}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.3

Replace all occurrences of $u_{2}$ with ${(9 - x)}^{\frac{1}{2}}$ .

$f'' (x) = - \frac{3 (\frac{2 {({(9 - x)}^{\frac{1}{2}})}^{2} + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4

Simplify.

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Step 2.19.3.4.1

Simplify each term.

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Step 2.19.3.4.1.1

Multiply the exponents in ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.19.3.4.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - \frac{3 (\frac{2 {(9 - x)}^{\frac{1}{2} \cdot 2} + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.1.1.2

Cancel the common factor of $2$ .

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Step 2.19.3.4.1.1.2.1

Cancel the common factor.

$f'' (x) = - \frac{3 (\frac{2 {(9 - x)}^{\frac{1}{2} \cdot 2} + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.1.1.2.2

Rewrite the expression.

$f'' (x) = - \frac{3 (\frac{2 (9 - x) + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.1.2

Simplify.

$f'' (x) = - \frac{3 (\frac{2 (9 - x) + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.1.3

Apply the distributive property.

$f'' (x) = - \frac{3 (\frac{2 \cdot 9 + 2 (- x) + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.1.4

Multiply $2$ by $9$ .

$f'' (x) = - \frac{3 (\frac{18 + 2 (- x) + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.1.5

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{3 (\frac{18 - 2 x + x - 6}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.2

Subtract $6$ from $18$ .

$f'' (x) = - \frac{3 (\frac{- 2 x + x + 12}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.3.4.3

Add $- 2 x$ and $x$ .

$f'' (x) = - \frac{3 (\frac{- x + 12}{2 {(9 - x)}^{\frac{1}{2}}})}{2 \cdot 9 + 2 (- x)}$

Step 2.19.4

Combine terms.

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Step 2.19.4.1

Combine $3$ and $\frac{- x + 12}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f'' (x) = - \frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{2 \cdot 9 + 2 (- x)}$

Step 2.19.4.2

Multiply $2$ by $9$ .

$f'' (x) = - \frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{18 + 2 (- x)}$

Step 2.19.4.3

Multiply $- 1$ by $2$ .

$f'' (x) = - \frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{18 - 2 x}$

Step 2.19.4.4

Rewrite $\frac{\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}}{18 - 2 x}$ as a product.

$f'' (x) = - (\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{1}{18 - 2 x})$

Step 2.19.4.5

Multiply $\frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}}}$ by $\frac{1}{18 - 2 x}$ .

$f'' (x) = - \frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}} (18 - 2 x)}$

Step 2.19.5

Simplify the denominator.

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Step 2.19.5.1

Factor $2$ out of $18 - 2 x$ .

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Step 2.19.5.1.1

Factor $2$ out of $18$ .

$f'' (x) = - \frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}} (2 (9) - 2 x)}$

Step 2.19.5.1.2

Factor $2$ out of $- 2 x$ .

$f'' (x) = - \frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}} (2 (9) + 2 (- x))}$

Step 2.19.5.1.3

Factor $2$ out of $2 (9) + 2 (- x)$ .

$f'' (x) = - \frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}} (2 (9 - x))}$

$f'' (x) = - \frac{3 (- x + 12)}{2 {(9 - x)}^{\frac{1}{2}} \cdot (2 (9 - x))}$

Step 2.19.5.2

Combine exponents.

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Step 2.19.5.2.1

Multiply $2$ by $2$ .

$f'' (x) = - \frac{3 (- x + 12)}{4 {(9 - x)}^{\frac{1}{2}} (9 - x)}$

Step 2.19.5.2.2

Raise $9 - x$ to the power of $1$ .

$f'' (x) = - \frac{3 (- x + 12)}{4 ((9 - x) {(9 - x)}^{\frac{1}{2}})}$

Step 2.19.5.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{3 (- x + 12)}{4 {(9 - x)}^{1 + \frac{1}{2}}}$

Step 2.19.5.2.4

Write $1$ as a fraction with a common denominator.

$f'' (x) = - \frac{3 (- x + 12)}{4 {(9 - x)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 2.19.5.2.5

Combine the numerators over the common denominator.

$f'' (x) = - \frac{3 (- x + 12)}{4 {(9 - x)}^{\frac{2 + 1}{2}}}$

Step 2.19.5.2.6

Add $2$ and $1$ .

$f'' (x) = - \frac{3 (- x + 12)}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 2.19.6

Factor $- 1$ out of $- x$ .

$f'' (x) = - \frac{3 (- (x) + 12)}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 2.19.7

Rewrite $12$ as $- 1 (- 12)$ .

$f'' (x) = - \frac{3 (- (x) - 1 \cdot - 12)}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 2.19.8

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

$f'' (x) = - \frac{3 (- (x - 12))}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 2.19.9

Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

$f'' (x) = - \frac{3 (- 1 (x - 12))}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 2.19.10

Move the negative in front of the fraction.

$f'' (x) = \frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 2.19.11

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}})$

Step 2.19.12

Multiply $\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}}$ by $1$ .

$f'' (x) = \frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}}$

Step 3

Find the third derivative.

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Step 3.1

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3 (x - 12)}{4 {(9 - x)}^{\frac{3}{2}}}$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [\frac{x - 12}{{(9 - x)}^{\frac{3}{2}}}]$ .

$f''' (x) = \frac{3}{4} \cdot \frac{d}{d x} (\frac{x - 12}{{(9 - x)}^{\frac{3}{2}}})$

Step 3.2

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} \frac{d}{d x} (x - 12) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{({(9 - x)}^{\frac{3}{2}})}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply the exponents in ${({(9 - x)}^{\frac{3}{2}})}^{2}$ .

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Step 3.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} \frac{d}{d x} (x - 12) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{\frac{3}{2} \cdot 2}}$

Step 3.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.2.1

Cancel the common factor.

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} \frac{d}{d x} (x - 12) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{\frac{3}{2} \cdot 2}}$

Step 3.3.1.2.2

Rewrite the expression.

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} \frac{d}{d x} (x - 12) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{3}}$

Step 3.3.2

By the Sum Rule, the derivative of $x - 12$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 12]$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} (\frac{d}{d x} (x) + \frac{d}{d x} (- 12)) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{3}}$

Step 3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} (1 + \frac{d}{d x} (- 12)) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{3}}$

Step 3.3.4

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} (1 + 0) - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{3}}$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} \cdot 1 - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{3}}$

Step 3.3.5.2

Multiply ${(9 - x)}^{\frac{3}{2}}$ by $1$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) \frac{d}{d x} {(9 - x)}^{\frac{3}{2}}}{{(9 - x)}^{3}}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 9 - x$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{1}$ as $9 - x$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{d}{d u (1)} ({u_{1}}^{\frac{3}{2}}) \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{3}{2}$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} {u_{1}}^{\frac{3}{2} - 1} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.4.3

Replace all occurrences of $u_{1}$ with $9 - x$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} \cdot {(9 - x)}^{\frac{3}{2} - 1} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} \cdot {(9 - x)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.6

Combine $- 1$ and $\frac{2}{2}$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} \cdot {(9 - x)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.7

Combine the numerators over the common denominator.

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} \cdot {(9 - x)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} \cdot {(9 - x)}^{\frac{3 - 2}{2}} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.8.2

Subtract $2$ from $3$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3}{2} \cdot {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.9

Combine $\frac{3}{2}$ and ${(9 - x)}^{\frac{1}{2}}$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 3.10

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot (\frac{d}{d x} (9) + \frac{d}{d x} (- x)))}{{(9 - x)}^{3}}$

Step 3.11

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot (0 + \frac{d}{d x} (- x)))}{{(9 - x)}^{3}}$

Step 3.12

Add $0$ and $\frac{d}{d x} [- x]$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot \frac{d}{d x} (- x))}{{(9 - x)}^{3}}$

Step 3.13

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} - (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot (- \frac{d}{d x} x))}{{(9 - x)}^{3}}$

Step 3.14

Multiply.

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Step 3.14.1

Multiply $- 1$ by $- 1$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} + 1 (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot (\frac{d}{d x} (x)))}{{(9 - x)}^{3}}$

Step 3.14.2

Multiply $x - 12$ by $1$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} + (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot (\frac{d}{d x} (x)))}{{(9 - x)}^{3}}$

Step 3.15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} + (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}) \cdot 1}{{(9 - x)}^{3}}$

Step 3.16

Combine fractions.

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Step 3.16.1

Multiply $x - 12$ by $1$ .

$f''' (x) = \frac{3}{4} \cdot \frac{{(9 - x)}^{\frac{3}{2}} + (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2})}{{(9 - x)}^{3}}$

Step 3.16.2

Multiply $\frac{3}{4}$ by $\frac{{(9 - x)}^{\frac{3}{2}} + (x - 12) \frac{3 {(9 - x)}^{\frac{1}{2}}}{2}}{{(9 - x)}^{3}}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}))}{4 {(9 - x)}^{3}}$

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

$f''' (x) = \frac{3 {(9 - x)}^{\frac{3}{2}} + 3 ((x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}))}{4 {(9 - x)}^{3}}$

Step 3.17.2

Simplify the numerator.

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Step 3.17.2.1

Factor $3$ out of $3 {(9 - x)}^{\frac{3}{2}} + 3 (x - 12) \frac{3 {(9 - x)}^{\frac{1}{2}}}{2}$ .

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Step 3.17.2.1.1

Factor $3$ out of $3 {(9 - x)}^{\frac{3}{2}}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}}) + 3 (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.1.2

Factor $3$ out of $3 (x - 12) \frac{3 {(9 - x)}^{\frac{1}{2}}}{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}}) + 3 ((x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}))}{4 {(9 - x)}^{3}}$

Step 3.17.2.1.3

Factor $3$ out of $3 ({(9 - x)}^{\frac{3}{2}}) + 3 ((x - 12) \frac{3 {(9 - x)}^{\frac{1}{2}}}{2})$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + (x - 12) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}))}{4 {(9 - x)}^{3}}$

Step 3.17.2.2

Apply the distributive property.

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + x (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}) - 12 \frac{3 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.3

Combine $x$ and $\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{x (3 {(9 - x)}^{\frac{1}{2}})}{2} - 12 \frac{3 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.4

Cancel the common factor of $2$ .

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Step 3.17.2.4.1

Factor $2$ out of $- 12$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{x (3 {(9 - x)}^{\frac{1}{2}})}{2} + 2 (- 6) (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}))}{4 {(9 - x)}^{3}}$

Step 3.17.2.4.2

Cancel the common factor.

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{x (3 {(9 - x)}^{\frac{1}{2}})}{2} + 2 \cdot (- 6 \frac{3 {(9 - x)}^{\frac{1}{2}}}{2}))}{4 {(9 - x)}^{3}}$

Step 3.17.2.4.3

Rewrite the expression.

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{x (3 {(9 - x)}^{\frac{1}{2}})}{2} - 6 (3 {(9 - x)}^{\frac{1}{2}}))}{4 {(9 - x)}^{3}}$

Step 3.17.2.5

Multiply $3$ by $- 6$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{x (3 {(9 - x)}^{\frac{1}{2}})}{2} - 18 {(9 - x)}^{\frac{1}{2}})}{4 {(9 - x)}^{3}}$

Step 3.17.2.6

Move $3$ to the left of $x$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 x {(9 - x)}^{\frac{1}{2}}}{2} - 18 {(9 - x)}^{\frac{1}{2}})}{4 {(9 - x)}^{3}}$

Step 3.17.2.7

To write $- 18 {(9 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 x {(9 - x)}^{\frac{1}{2}}}{2} - 18 {(9 - x)}^{\frac{1}{2}} \cdot \frac{2}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.8

Combine $- 18 {(9 - x)}^{\frac{1}{2}}$ and $\frac{2}{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 x {(9 - x)}^{\frac{1}{2}}}{2} + \frac{- 18 {(9 - x)}^{\frac{1}{2}} \cdot 2}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.9

Combine the numerators over the common denominator.

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 x {(9 - x)}^{\frac{1}{2}} - 18 {(9 - x)}^{\frac{1}{2}} \cdot 2}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.10

Simplify the numerator.

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Step 3.17.2.10.1

Multiply $2$ by $- 18$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 x {(9 - x)}^{\frac{1}{2}} - 36 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.10.2

Rewrite $3 x {(9 - x)}^{\frac{1}{2}} - 36 {(9 - x)}^{\frac{1}{2}}$ in a factored form.

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Step 3.17.2.10.2.1

Add parentheses.

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 (x {(9 - x)}^{\frac{1}{2}}) - 36 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.10.2.2

Factor $3 u_{2}$ out of $3 x u_{2} - 36 u_{2}$ .

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Step 3.17.2.10.2.2.1

Factor $3 u_{2}$ out of $3 x u_{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 u_{2} (x) - 36 u_{2}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.10.2.2.2

Factor $3 u_{2}$ out of $- 36 u_{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 u_{2} (x) + 3 u_{2} (- 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.10.2.2.3

Factor $3 u_{2}$ out of $3 u_{2} (x) + 3 u_{2} (- 12)$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 u_{2} (x - 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.10.2.3

Replace all occurrences of $u_{2}$ with ${(9 - x)}^{\frac{1}{2}}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} + \frac{3 {(9 - x)}^{\frac{1}{2}} (x - 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.11

To write ${(9 - x)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f''' (x) = \frac{3 ({(9 - x)}^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{3 {(9 - x)}^{\frac{1}{2}} (x - 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.12

Combine ${(9 - x)}^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$f''' (x) = \frac{3 (\frac{{(9 - x)}^{\frac{3}{2}} \cdot 2}{2} + \frac{3 {(9 - x)}^{\frac{1}{2}} (x - 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.13

Combine the numerators over the common denominator.

$f''' (x) = \frac{3 (\frac{{(9 - x)}^{\frac{3}{2}} \cdot 2 + 3 {(9 - x)}^{\frac{1}{2}} (x - 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.14

Rewrite $\frac{{(9 - x)}^{\frac{3}{2}} \cdot 2 + 3 {(9 - x)}^{\frac{1}{2}} (x - 12)}{2}$ in a factored form.

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Step 3.17.2.14.1

Move $2$ to the left of ${(9 - x)}^{\frac{3}{2}}$ .

$f''' (x) = \frac{3 (\frac{2 \cdot {(9 - x)}^{\frac{3}{2}} + 3 {(9 - x)}^{\frac{1}{2}} (x - 12)}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.14.2

Apply the distributive property.

$f''' (x) = \frac{3 (\frac{2 {(9 - x)}^{\frac{3}{2}} + 3 {(9 - x)}^{\frac{1}{2}} x + 3 {(9 - x)}^{\frac{1}{2}} \cdot - 12}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.14.3

Multiply $- 12$ by $3$ .

$f''' (x) = \frac{3 (\frac{2 {(9 - x)}^{\frac{3}{2}} + 3 {(9 - x)}^{\frac{1}{2}} x - 36 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.2.14.4

Reorder terms.

$f''' (x) = \frac{3 (\frac{3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}}{2})}{4 {(9 - x)}^{3}}$

Step 3.17.3

Combine terms.

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Step 3.17.3.1

Combine $3$ and $\frac{3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}}{2}$ .

$f''' (x) = \frac{\frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{2}}{4 {(9 - x)}^{3}}$

Step 3.17.3.2

Rewrite $\frac{\frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{2}}{4 {(9 - x)}^{3}}$ as a product.

$f''' (x) = \frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{2} \cdot \frac{1}{4 {(9 - x)}^{3}}$

Step 3.17.3.3

Multiply $\frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{2}$ by $\frac{1}{4 {(9 - x)}^{3}}$ .

$f''' (x) = \frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{2 (4 {(9 - x)}^{3})}$

Step 3.17.3.4

Multiply $4$ by $2$ .

$f''' (x) = \frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{8 {(9 - x)}^{3}}$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\frac{3}{8}$ is constant with respect to $x$ , the derivative of $\frac{3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}})}{8 {(9 - x)}^{3}}$ with respect to $x$ is $\frac{3}{8} \frac{d}{d x} [\frac{3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}}{{(9 - x)}^{3}}]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{d}{d x} (\frac{3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}}{{(9 - x)}^{3}})$

Step 4.2

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} \frac{d}{d x} (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{({(9 - x)}^{3})}^{2}}$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply the exponents in ${({(9 - x)}^{3})}^{2}$ .

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Step 4.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

Step 4.3.1.2

Multiply $3$ by $2$ .

Step 4.3.2

By the Sum Rule, the derivative of $3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [3 x {(9 - x)}^{\frac{1}{2}}] + \frac{d}{d x} [2 {(9 - x)}^{\frac{3}{2}}] + \frac{d}{d x} [- 36 {(9 - x)}^{\frac{1}{2}}]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{d}{d x} (3 x {(9 - x)}^{\frac{1}{2}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x {(9 - x)}^{\frac{1}{2}}$ with respect to $x$ is $3 \frac{d}{d x} [x {(9 - x)}^{\frac{1}{2}}]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 \frac{d}{d x} (x {(9 - x)}^{\frac{1}{2}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.4

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x \frac{d}{d x} ({(9 - x)}^{\frac{1}{2}}) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 - x$ .

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Step 4.5.1

To apply the Chain Rule, set $u_{1}$ as $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.5.3

Replace all occurrences of $u_{1}$ with $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.7

Combine $- 1$ and $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.8

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Multiply $- 1$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.9.2

Subtract $2$ from $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{\frac{- 1}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.10

Combine fractions.

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Step 4.10.1

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2} \cdot {(9 - x)}^{- \frac{1}{2}} \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.10.2

Combine $\frac{1}{2}$ and ${(9 - x)}^{- \frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{{(9 - x)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.10.3

Move ${(9 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (x (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.10.4

Combine $\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ and $x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.11

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (9) + \frac{d}{d x} (- x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.12

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x)) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.13

Add $0$ and $\frac{d}{d x} [- x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.14

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.15

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot (- 1 \cdot 1) + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.16

Combine fractions.

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Step 4.16.1

Multiply $- 1$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x}{2 {(9 - x)}^{\frac{1}{2}}} \cdot - 1 + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.16.2

Combine $\frac{x}{2 {(9 - x)}^{\frac{1}{2}}}$ and $- 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{x \cdot - 1}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.16.3

Simplify the expression.

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Step 4.16.3.1

Move $- 1$ to the left of $x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- 1 \cdot x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.16.3.2

Rewrite $- 1 x$ as $- x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.16.3.3

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (- \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (x)) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (- \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \cdot 1) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.18

Multiply ${(9 - x)}^{\frac{1}{2}}$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (- \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.19

To write ${(9 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (- \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + {(9 - x)}^{\frac{1}{2}} \cdot \frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.20

Combine ${(9 - x)}^{\frac{1}{2}}$ and $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (- \frac{x}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{{(9 - x)}^{\frac{1}{2}} (2 {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.21

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + {(9 - x)}^{\frac{1}{2}} (2 {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.22

Multiply ${(9 - x)}^{\frac{1}{2}}$ by ${(9 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.22.1

Move ${(9 - x)}^{\frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{\frac{1}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.22.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + {(9 - x)}^{\frac{1}{2} + \frac{1}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.22.3

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + {(9 - x)}^{\frac{1 + 1}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.22.4

Add $1$ and $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + {(9 - x)}^{\frac{2}{2}} \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.22.5

Divide $2$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + (9 - x) \cdot 2}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.23

Differentiate using the Constant Multiple Rule.

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Step 4.23.1

Simplify ${(9 - x)}^{1} \cdot 2$ .

Step 4.23.2

Combine fractions.

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Step 4.23.2.1

Move $2$ to the left of $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (3 (\frac{- x + 2 \cdot (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}}) + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.23.2.2

Combine $3$ and $\frac{- x + 2 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (2 {(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.23.3

Since $2$ is constant with respect to $x$ , the derivative of $2 {(9 - x)}^{\frac{3}{2}}$ with respect to $x$ is $2 \frac{d}{d x} [{(9 - x)}^{\frac{3}{2}}]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 \frac{d}{d x} ({(9 - x)}^{\frac{3}{2}}) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.24

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 9 - x$ .

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Step 4.24.1

To apply the Chain Rule, set $u_{2}$ as $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{d}{d u (2)} ({u_{2}}^{\frac{3}{2}}) \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.24.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{3}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} {u_{2}}^{\frac{3}{2} - 1} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.24.3

Replace all occurrences of $u_{2}$ with $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} \cdot {(9 - x)}^{\frac{3}{2} - 1} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.25

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} \cdot {(9 - x)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.26

Combine $- 1$ and $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} \cdot {(9 - x)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.27

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} \cdot {(9 - x)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.28

Simplify the numerator.

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Step 4.28.1

Multiply $- 1$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} \cdot {(9 - x)}^{\frac{3 - 2}{2}} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.28.2

Subtract $2$ from $3$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3}{2} \cdot {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.29

Combine $\frac{3}{2}$ and ${(9 - x)}^{\frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 2 (\frac{3 {(9 - x)}^{\frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.30

Combine $\frac{3 {(9 - x)}^{\frac{1}{2}}}{2}$ and $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{3 {(9 - x)}^{\frac{1}{2}} \cdot 2}{2} \cdot \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.31

Multiply $2$ by $3$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{6 {(9 - x)}^{\frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.32

Factor $2$ out of $6 {(9 - x)}^{\frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{2 (3 {(9 - x)}^{\frac{1}{2}})}{2} \cdot \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.33

Cancel the common factors.

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Step 4.33.1

Factor $2$ out of $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{2 (3 {(9 - x)}^{\frac{1}{2}})}{2 (1)} \cdot \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.33.2

Cancel the common factor.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{2 (3 {(9 - x)}^{\frac{1}{2}})}{2 \cdot 1} \cdot \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.33.3

Rewrite the expression.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{3 {(9 - x)}^{\frac{1}{2}}}{1} \cdot \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.33.4

Divide $3 {(9 - x)}^{\frac{1}{2}}$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 3 {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (9 - x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.34

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 3 {(9 - x)}^{\frac{1}{2}} (\frac{d}{d x} (9) + \frac{d}{d x} (- x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.35

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 3 {(9 - x)}^{\frac{1}{2}} (0 + \frac{d}{d x} (- x)) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.36

Add $0$ and $\frac{d}{d x} [- x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 3 {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} (- x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.37

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + 3 {(9 - x)}^{\frac{1}{2}} (- \frac{d}{d x} x) + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.38

Multiply $- 1$ by $3$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} - 3 {(9 - x)}^{\frac{1}{2}} \frac{d}{d x} x + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.39

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} - 3 {(9 - x)}^{\frac{1}{2}} \cdot 1 + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.40

Multiply $- 3$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} - 3 {(9 - x)}^{\frac{1}{2}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.41

To write $- 3 {(9 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} - 3 {(9 - x)}^{\frac{1}{2}} \cdot \frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.42

Combine $- 3 {(9 - x)}^{\frac{1}{2}}$ and $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x))}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{- 3 {(9 - x)}^{\frac{1}{2}} (2 {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.43

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 3 {(9 - x)}^{\frac{1}{2}} (2 {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.44

Multiply $2$ by $- 3$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.45

Multiply ${(9 - x)}^{\frac{1}{2}}$ by ${(9 - x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.45.1

Move ${(9 - x)}^{\frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 ({(9 - x)}^{\frac{1}{2}} {(9 - x)}^{\frac{1}{2}})}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.45.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 {(9 - x)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.45.3

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 {(9 - x)}^{\frac{1 + 1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.45.4

Add $1$ and $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 {(9 - x)}^{\frac{2}{2}}}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.45.5

Divide $2$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{d}{d x} (- 36 {(9 - x)}^{\frac{1}{2}})) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.46

Differentiate using the Constant Multiple Rule.

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Step 4.46.1

Simplify $- 6 {(9 - x)}^{1}$ .

Step 4.46.2

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36 {(9 - x)}^{\frac{1}{2}}$ with respect to $x$ is $- 36 \frac{d}{d x} [{(9 - x)}^{\frac{1}{2}}]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 \frac{d}{d x} {(9 - x)}^{\frac{1}{2}}) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.47

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 9 - x$ .

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Step 4.47.1

To apply the Chain Rule, set $u_{3}$ as $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{d}{d u (3)} ({u_{3}}^{\frac{1}{2}}) \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.47.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = \frac{1}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} {u_{3}}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.47.3

Replace all occurrences of $u_{3}$ with $9 - x$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.48

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.49

Combine $- 1$ and $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.50

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.51

Simplify the numerator.

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Step 4.51.1

Multiply $- 1$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.51.2

Subtract $2$ from $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{\frac{- 1}{2}} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.52

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2} \cdot {(9 - x)}^{- \frac{1}{2}} \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.53

Combine $\frac{1}{2}$ and ${(9 - x)}^{- \frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{{(9 - x)}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.54

Move ${(9 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - 36 (\frac{1}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.55

Combine $\frac{1}{2 {(9 - x)}^{\frac{1}{2}}}$ and $- 36$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{- 36}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.56

Factor $2$ out of $- 36$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{2 \cdot - 18}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.57

Cancel the common factors.

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Step 4.57.1

Factor $2$ out of $2 {(9 - x)}^{\frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{2 \cdot - 18}{2 ({(9 - x)}^{\frac{1}{2}})} \cdot \frac{d}{d x} (9 - x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.57.2

Cancel the common factor.

Step 4.57.3

Rewrite the expression.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{- 18}{{(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58

Differentiate.

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Step 4.58.1

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (9 - x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.2

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (9) + \frac{d}{d x} (- x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x))) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.4

Add $0$ and $\frac{d}{d x} [- x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} - \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.6

Multiply.

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Step 4.58.6.1

Multiply $- 1$ by $- 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + 1 (\frac{18}{{(9 - x)}^{\frac{1}{2}}}) \cdot \frac{d}{d x} (x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.6.2

Multiply $\frac{18}{{(9 - x)}^{\frac{1}{2}}}$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x)) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} \cdot 1) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.58.8

Multiply $\frac{18}{{(9 - x)}^{\frac{1}{2}}}$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) \frac{d}{d x} {(9 - x)}^{3}}{{(9 - x)}^{6}}$

Step 4.59

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 9 - x$ .

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Step 4.59.1

To apply the Chain Rule, set $u_{4}$ as $9 - x$ .

Step 4.59.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = 3$ .

Step 4.59.3

Replace all occurrences of $u_{4}$ with $9 - x$ .

Step 4.60

Simplify with factoring out.

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Step 4.60.1

Multiply $3$ by $- 1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x))}{{(9 - x)}^{6}}$

Step 4.60.2

Factor ${(9 - x)}^{2}$ out of ${(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x])$ .

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Step 4.60.2.1

Factor ${(9 - x)}^{2}$ out of ${(9 - x)}^{3} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}})$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}})) - 3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x))}{{(9 - x)}^{6}}$

Step 4.60.2.2

Factor ${(9 - x)}^{2}$ out of $- 3 (3 x {(9 - x)}^{\frac{1}{2}} + 2 {(9 - x)}^{\frac{3}{2}} - 36 {(9 - x)}^{\frac{1}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x])$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}})) + {(9 - x)}^{2} (- 3 (3 x {(9 - x)}^{\frac{- 3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{\frac{- 3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{6}}$

Step 4.60.2.3

Factor ${(9 - x)}^{2}$ out of ${(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}})) + {(9 - x)}^{2} (- 3 (3 x {(9 - x)}^{\frac{- 3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{\frac{- 3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x]))$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{\frac{- 3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{\frac{- 3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{6}}$

Step 4.60.3

Move the negative in front of fractions.

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Step 4.60.3.1

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{\frac{- 3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{6}}$

Step 4.60.3.2

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{- \frac{3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{6}}$

Step 4.61

Cancel the common factors.

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Step 4.61.1

Factor ${(9 - x)}^{2}$ out of ${(9 - x)}^{6}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{- \frac{3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{2} {(9 - x)}^{4}}$

Step 4.61.2

Cancel the common factor.

$f^{4} (x) = \frac{3}{8} \cdot \frac{{(9 - x)}^{2} ((9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{- \frac{3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{2} {(9 - x)}^{4}}$

Step 4.61.3

Rewrite the expression.

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{- \frac{3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x))}{{(9 - x)}^{4}}$

Step 4.62

Factor $9 - x$ out of $(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) - 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{- \frac{3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x])$ .

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Step 4.62.1

Factor $9 - x$ out of $- 3 (3 x {(9 - x)}^{- \frac{3}{2}} + 2 {(9 - x)}^{- \frac{1}{2}} - 36 {(9 - x)}^{- \frac{3}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x])$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) + (9 - x) (- 3 (3 x {(9 - x)}^{\frac{- 5}{2}} + 2 {(9 - x)}^{\frac{- 3}{2}} - 36 {(9 - x)}^{\frac{- 5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{4}}$

Step 4.62.2

Factor $9 - x$ out of $(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}}) + (9 - x) (- 3 (3 x {(9 - x)}^{\frac{- 5}{2}} + 2 {(9 - x)}^{\frac{- 3}{2}} - 36 {(9 - x)}^{\frac{- 5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x]))$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{\frac{- 5}{2}} + 2 {(9 - x)}^{\frac{- 3}{2}} - 36 {(9 - x)}^{\frac{- 5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{4}}$

Step 4.63

Move the negative in front of fractions.

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Step 4.63.1

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{\frac{- 3}{2}} - 36 {(9 - x)}^{\frac{- 5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{4}}$

Step 4.63.2

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{\frac{- 5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{4}}$

Step 4.63.3

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{{(9 - x)}^{4}}$

Step 4.64

Cancel the common factors.

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Step 4.64.1

Factor $9 - x$ out of ${(9 - x)}^{4}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{(9 - x) {(9 - x)}^{3}}$

Step 4.64.2

Cancel the common factor.

$f^{4} (x) = \frac{3}{8} \cdot \frac{(9 - x) (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{(9 - x) {(9 - x)}^{3}}$

Step 4.64.3

Rewrite the expression.

$f^{4} (x) = \frac{3}{8} \cdot \frac{\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}}$

Step 4.65

Multiply $\frac{\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} [9 - x])}{{(9 - x)}^{3}}$ by $\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}}$ .

$f^{4} (x) = \frac{3}{8} \cdot (\frac{2 {(9 - x)}^{\frac{1}{2}}}{2 {(9 - x)}^{\frac{1}{2}}} \cdot \frac{\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x))}{{(9 - x)}^{3}})$

Step 4.66

Combine.

$f^{4} (x) = \frac{3}{8} \cdot \frac{2 {(9 - x)}^{\frac{1}{2}} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}} + \frac{18}{{(9 - x)}^{\frac{1}{2}}} - 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.67

Apply the distributive property.

$f^{4} (x) = \frac{3}{8} \cdot \frac{2 {(9 - x)}^{\frac{1}{2}} (\frac{3 (- x + 2 (9 - x)) - 6 (9 - x)}{2 {(9 - x)}^{\frac{1}{2}}}) + 2 {(9 - x)}^{\frac{1}{2}} (\frac{18}{{(9 - x)}^{\frac{1}{2}}}) + 2 {(9 - x)}^{\frac{1}{2}} (- 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.68

Cancel the common factor of $2 {(9 - x)}^{\frac{1}{2}}$ .

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Step 4.68.1

Cancel the common factor.

Step 4.68.2

Rewrite the expression.

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 2 {(9 - x)}^{\frac{1}{2}} (\frac{18}{{(9 - x)}^{\frac{1}{2}}}) + 2 {(9 - x)}^{\frac{1}{2}} (- 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.69

Cancel the common factor of ${(9 - x)}^{\frac{1}{2}}$ .

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Step 4.69.1

Factor ${(9 - x)}^{\frac{1}{2}}$ out of $2 {(9 - x)}^{\frac{1}{2}}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + {(9 - x)}^{\frac{1}{2}} \cdot (2 (\frac{18}{{(9 - x)}^{\frac{1}{2}}})) + 2 {(9 - x)}^{\frac{1}{2}} (- 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.69.2

Cancel the common factor.

Step 4.69.3

Rewrite the expression.

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 2 \cdot 18 + 2 {(9 - x)}^{\frac{1}{2}} (- 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.70

Multiply.

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Step 4.70.1

Multiply $2$ by $18$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 2 {(9 - x)}^{\frac{1}{2}} (- 3 (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.70.2

Multiply $- 3$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{\frac{1}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) ({(9 - x)}^{2} \frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.71

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{2 + \frac{1}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.72

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{2 \cdot \frac{2}{2} + \frac{1}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.73

Combine $2$ and $\frac{2}{2}$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{\frac{2 \cdot 2}{2} + \frac{1}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.74

Combine the numerators over the common denominator.

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{\frac{2 \cdot 2 + 1}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.75

Simplify the numerator.

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Step 4.75.1

Multiply $2$ by $2$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{\frac{4 + 1}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.75.2

Add $4$ and $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{\frac{5}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (9 - x)))}{2 {(9 - x)}^{\frac{1}{2}} {(9 - x)}^{3}}$

Step 4.76

Multiply ${(9 - x)}^{\frac{1}{2}}$ by ${(9 - x)}^{3}$ by adding the exponents.

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Step 4.76.1

Move ${(9 - x)}^{3}$ .

Step 4.76.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 4.76.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

Step 4.76.4

Combine $3$ and $\frac{2}{2}$ .

Step 4.76.5

Combine the numerators over the common denominator.

Step 4.76.6

Simplify the numerator.

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Step 4.76.6.1

Multiply $3$ by $2$ .

Step 4.76.6.2

Add $6$ and $1$ .

Step 4.77

By the Sum Rule, the derivative of $9 - x$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x]$ .

Step 4.78

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 - 6 {(9 - x)}^{\frac{5}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (0 + \frac{d}{d x} (- x)))}{2 {(9 - x)}^{\frac{7}{2}}}$

Step 4.79

Add $0$ and $\frac{d}{d x} [- x]$ .

Step 4.80

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

Step 4.81

Multiply $- 1$ by $- 6$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} ((3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}) (\frac{d}{d x} (x)))}{2 {(9 - x)}^{\frac{7}{2}}}$

Step 4.82

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

Step 4.83

Combine fractions.

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Step 4.83.1

Multiply $3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}$ by $1$ .

$f^{4} (x) = \frac{3}{8} \cdot \frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}})}{2 {(9 - x)}^{\frac{7}{2}}}$

Step 4.83.2

Multiply $\frac{3}{8}$ by $\frac{3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}})}{2 {(9 - x)}^{\frac{7}{2}}}$ .

$f^{4} (x) = \frac{3 (3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}))}{8 (2 {(9 - x)}^{\frac{7}{2}})}$

Step 4.83.3

Multiply $2$ by $8$ .

$f^{4} (x) = \frac{3 (3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x {(9 - x)}^{- \frac{5}{2}} + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84

Simplify.

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Step 4.84.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = \frac{3 (3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 {(9 - x)}^{- \frac{3}{2}} - 36 {(9 - x)}^{- \frac{5}{2}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = \frac{3 (3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 {(9 - x)}^{- \frac{5}{2}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f^{4} (x) = \frac{3 (3 (- x + 2 (9 - x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.4

Apply the distributive property.

$f^{4} (x) = \frac{3 (3 (- x + 2 \cdot 9 + 2 (- x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.5

Apply the distributive property.

$f^{4} (x) = \frac{3 (3 (- x) + 3 (2 \cdot 9) + 3 (2 (- x)) - 6 (9 - x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.6

Apply the distributive property.

$f^{4} (x) = \frac{3 (3 (- x) + 3 (2 \cdot 9) + 3 (2 (- x)) - 6 \cdot 9 - 6 (- x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.7

Apply the distributive property.

$f^{4} (x) = \frac{3 (3 (- x)) + 3 (3 (2 \cdot 9)) + 3 (3 (2 (- x))) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 \cdot 36 + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8

Simplify the numerator.

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Step 4.84.8.1

Factor $3$ out of $3 \cdot 3 (- x) + 3 \cdot 3 \cdot 2 \cdot 9 + 3 \cdot 3 \cdot 2 (- x) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 \cdot 36 + 3 \cdot 6 {(9 - x)}^{\frac{5}{2}} (3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}} + 2 \frac{1}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})$ .

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Step 4.84.8.1.1

Factor $3$ out of $3 \cdot 3 (- x)$ .

$f^{4} (x) = \frac{3 (3 (- x)) + 3 \cdot 3 \cdot 2 \cdot 9 + 3 \cdot 3 \cdot (2 (- x)) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 \cdot 36 + 3 \cdot (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.2

Factor $3$ out of $3 \cdot 3 \cdot 2 \cdot 9$ .

$f^{4} (x) = \frac{3 (3 (- x)) + 3 (3 \cdot 2 \cdot 9) + 3 \cdot 3 \cdot (2 (- x)) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 \cdot 36 + 3 \cdot (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.3

Factor $3$ out of $3 \cdot 3 \cdot 2 (- x)$ .

$f^{4} (x) = \frac{3 (3 (- x)) + 3 (3 \cdot 2 \cdot 9) + 3 (3 \cdot (2 (- x))) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 \cdot 36 + 3 \cdot (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.4

Factor $3$ out of $3 \cdot 36$ .

$f^{4} (x) = \frac{3 (3 (- x)) + 3 (3 \cdot 2 \cdot 9) + 3 (3 \cdot (2 (- x))) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 (36) + 3 \cdot (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.5

Factor $3$ out of $3 \cdot 6 {(9 - x)}^{\frac{5}{2}} (3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}} + 2 \frac{1}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})$ .

$f^{4} (x) = \frac{3 (3 (- x)) + 3 (3 \cdot 2 \cdot 9) + 3 (3 \cdot (2 (- x))) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 (36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.6

Factor $3$ out of $3 (3 (- x)) + 3 (3 \cdot 2 \cdot 9)$ .

$f^{4} (x) = \frac{3 (3 (- x) + 3 \cdot 2 \cdot 9) + 3 (3 \cdot (2 (- x))) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 (36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.7

Factor $3$ out of $3 (3 (- x) + 3 \cdot 2 \cdot 9) + 3 (3 \cdot 2 (- x))$ .

$f^{4} (x) = \frac{3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x))) + 3 (- 6 \cdot 9) + 3 (- 6 (- x)) + 3 (36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.8

Factor $3$ out of $3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot 2 (- x)) + 3 (- 6 \cdot 9)$ .

$f^{4} (x) = \frac{3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x)) - 6 \cdot 9) + 3 (- 6 (- x)) + 3 (36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.9

Factor $3$ out of $3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot 2 (- x) - 6 \cdot 9) + 3 (- 6 (- x))$ .

$f^{4} (x) = \frac{3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x)) - 6 \cdot 9 - 6 (- x)) + 3 (36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.10

Factor $3$ out of $3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot 2 (- x) - 6 \cdot 9 - 6 (- x)) + 3 (36)$ .

$f^{4} (x) = \frac{3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x)) - 6 \cdot 9 - 6 (- x) + 36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.1.11

Factor $3$ out of $3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot 2 (- x) - 6 \cdot 9 - 6 (- x) + 36) + 3 (6 {(9 - x)}^{\frac{5}{2}} (3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}} + 2 \frac{1}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))$ .

$f^{4} (x) = \frac{3 (3 (- x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x)) - 6 \cdot 9 - 6 (- x) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.2

Subtract $6 (- x)$ from $3 (- x)$ .

$f^{4} (x) = \frac{3 (- 3 \cdot (- 1 x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x)) - 6 \cdot 9 + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3

Factor $3$ out of $- 3 \cdot - 1 x + 3 \cdot 2 \cdot 9 + 3 \cdot 2 (- x) - 6 \cdot 9 + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}} + 2 \frac{1}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})$ .

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Step 4.84.8.3.1

Factor $3$ out of $- 3 \cdot - 1 x$ .

$f^{4} (x) = \frac{3 (3 (1 x) + 3 \cdot 2 \cdot 9 + 3 \cdot (2 (- x)) - 6 \cdot 9 + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.2

Factor $3$ out of $3 \cdot 2 \cdot 9$ .

$f^{4} (x) = \frac{3 (3 (1 x) + 3 (2 \cdot 9) + 3 \cdot (2 (- x)) - 6 \cdot 9 + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.3

Factor $3$ out of $3 \cdot 2 (- x)$ .

$f^{4} (x) = \frac{3 (3 (1 x) + 3 (2 \cdot 9) + 3 (2 (- x)) - 6 \cdot 9 + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.4

Factor $3$ out of $- 6 \cdot 9$ .

$f^{4} (x) = \frac{3 (3 (1 x) + 3 (2 \cdot 9) + 3 (2 (- x)) + 3 (- 2 \cdot 9) + 36 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.5

Factor $3$ out of $36$ .

$f^{4} (x) = \frac{3 (3 (1 x) + 3 (2 \cdot 9) + 3 (2 (- x)) + 3 (- 2 \cdot 9) + 3 \cdot 12 + 6 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.6

Factor $3$ out of $6 {(9 - x)}^{\frac{5}{2}} (3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}} + 2 \frac{1}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})$ .

$f^{4} (x) = \frac{3 (3 (1 x) + 3 (2 \cdot 9) + 3 (2 (- x)) + 3 (- 2 \cdot 9) + 3 \cdot 12 + 3 (2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.7

Factor $3$ out of $3 (- - 1 x) + 3 (2 \cdot 9)$ .

$f^{4} (x) = \frac{3 (3 (1 x + 2 \cdot 9) + 3 (2 (- x)) + 3 (- 2 \cdot 9) + 3 \cdot 12 + 3 (2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.8

Factor $3$ out of $3 (- - 1 x + 2 \cdot 9) + 3 (2 (- x))$ .

$f^{4} (x) = \frac{3 (3 (1 x + 2 \cdot 9 + 2 (- x)) + 3 (- 2 \cdot 9) + 3 \cdot 12 + 3 (2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.9

Factor $3$ out of $3 (- - 1 x + 2 \cdot 9 + 2 (- x)) + 3 (- 2 \cdot 9)$ .

$f^{4} (x) = \frac{3 (3 (1 x + 2 \cdot 9 + 2 (- x) - 2 \cdot 9) + 3 \cdot 12 + 3 (2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.10

Factor $3$ out of $3 (- - 1 x + 2 \cdot 9 + 2 (- x) - 2 \cdot 9) + 3 \cdot 12$ .

$f^{4} (x) = \frac{3 (3 (1 x + 2 \cdot 9 + 2 (- x) - 2 \cdot 9 + 12) + 3 (2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.3.11

Factor $3$ out of $3 (- - 1 x + 2 \cdot 9 + 2 (- x) - 2 \cdot 9 + 12) + 3 (2 {(9 - x)}^{\frac{5}{2}} (3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}} + 2 \frac{1}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}}))$ .

$f^{4} (x) = \frac{3 (3 (1 x + 2 \cdot 9 + 2 (- x) - 2 \cdot 9 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.4

Multiply $- 1$ by $- 1$ .

Step 4.84.8.5

Multiply $x$ by $1$ .

$f^{4} (x) = \frac{3 (3 (x + 2 \cdot 9 + 2 (- x) - 2 \cdot 9 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.6

Multiply $2$ by $9$ .

$f^{4} (x) = \frac{3 (3 (x + 18 + 2 (- x) - 2 \cdot 9 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.7

Multiply $- 1$ by $2$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 2 \cdot 9 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.8

Multiply $- 2$ by $9$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (3 x (\frac{1}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.9

Simplify each term.

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Step 4.84.8.9.1

Multiply $3 x \frac{1}{{(9 - x)}^{\frac{5}{2}}}$ .

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Step 4.84.8.9.1.1

Combine $3$ and $\frac{1}{{(9 - x)}^{\frac{5}{2}}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (x (\frac{3}{{(9 - x)}^{\frac{5}{2}}}) + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.9.1.2

Combine $x$ and $\frac{3}{{(9 - x)}^{\frac{5}{2}}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (\frac{x \cdot 3}{{(9 - x)}^{\frac{5}{2}}} + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.9.2

Move $3$ to the left of $x$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (\frac{3 \cdot x}{{(9 - x)}^{\frac{5}{2}}} + 2 (\frac{1}{{(9 - x)}^{\frac{3}{2}}}) - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.9.3

Combine $2$ and $\frac{1}{{(9 - x)}^{\frac{3}{2}}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (\frac{3 x}{{(9 - x)}^{\frac{5}{2}}} + \frac{2}{{(9 - x)}^{\frac{3}{2}}} - 36 \frac{1}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.9.4

Combine $- 36$ and $\frac{1}{{(9 - x)}^{\frac{5}{2}}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (\frac{3 x}{{(9 - x)}^{\frac{5}{2}}} + \frac{2}{{(9 - x)}^{\frac{3}{2}}} + \frac{- 36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.9.5

Move the negative in front of the fraction.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (\frac{3 x}{{(9 - x)}^{\frac{5}{2}}} + \frac{2}{{(9 - x)}^{\frac{3}{2}}} - \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.10

Apply the distributive property.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 {(9 - x)}^{\frac{5}{2}} (\frac{3 x}{{(9 - x)}^{\frac{5}{2}}}) + 2 {(9 - x)}^{\frac{5}{2}} (\frac{2}{{(9 - x)}^{\frac{3}{2}}}) + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11

Simplify.

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Step 4.84.8.11.1

Cancel the common factor of ${(9 - x)}^{\frac{5}{2}}$ .

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Step 4.84.8.11.1.1

Factor ${(9 - x)}^{\frac{5}{2}}$ out of $2 {(9 - x)}^{\frac{5}{2}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + {(9 - x)}^{\frac{5}{2}} \cdot (2 (\frac{3 x}{{(9 - x)}^{\frac{5}{2}}})) + 2 {(9 - x)}^{\frac{5}{2}} (\frac{2}{{(9 - x)}^{\frac{3}{2}}}) + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.1.2

Cancel the common factor.

Step 4.84.8.11.1.3

Rewrite the expression.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 (3 x) + 2 {(9 - x)}^{\frac{5}{2}} (\frac{2}{{(9 - x)}^{\frac{3}{2}}}) + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.2

Multiply $3$ by $2$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 2 {(9 - x)}^{\frac{5}{2}} (\frac{2}{{(9 - x)}^{\frac{3}{2}}}) + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.3

Cancel the common factor of ${(9 - x)}^{\frac{3}{2}}$ .

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Step 4.84.8.11.3.1

Factor ${(9 - x)}^{\frac{3}{2}}$ out of $2 {(9 - x)}^{\frac{5}{2}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + {(9 - x)}^{\frac{3}{2}} (2 {(9 - x)}^{\frac{2}{2}}) (\frac{2}{{(9 - x)}^{\frac{3}{2}}}) + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.3.2

Cancel the common factor.

Step 4.84.8.11.3.3

Rewrite the expression.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 2 {(9 - x)}^{\frac{2}{2}} \cdot 2 + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.4

Multiply $2$ by $2$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 {(9 - x)}^{\frac{2}{2}} + 2 {(9 - x)}^{\frac{5}{2}} (- \frac{36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.5

Cancel the common factor of ${(9 - x)}^{\frac{5}{2}}$ .

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Step 4.84.8.11.5.1

Move the leading negative in $- \frac{36}{{(9 - x)}^{\frac{5}{2}}}$ into the numerator.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 {(9 - x)}^{\frac{2}{2}} + 2 {(9 - x)}^{\frac{5}{2}} (\frac{- 36}{{(9 - x)}^{\frac{5}{2}}})))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.5.2

Factor ${(9 - x)}^{\frac{5}{2}}$ out of $2 {(9 - x)}^{\frac{5}{2}}$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 {(9 - x)}^{\frac{2}{2}} + {(9 - x)}^{\frac{5}{2}} \cdot (2 (\frac{- 36}{{(9 - x)}^{\frac{5}{2}}}))))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.5.3

Cancel the common factor.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 {(9 - x)}^{\frac{2}{2}} + {(9 - x)}^{\frac{5}{2}} \cdot (2 (\frac{- 36}{{(9 - x)}^{\frac{5}{2}}}))))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.5.4

Rewrite the expression.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 {(9 - x)}^{\frac{2}{2}} + 2 \cdot - 36))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.11.6

Multiply $2$ by $- 36$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 {(9 - x)}^{\frac{2}{2}} - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.12

Simplify each term.

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Step 4.84.8.12.1

Divide $2$ by $2$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 (9 - x) - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.12.2

Simplify.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 (9 - x) - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.12.3

Apply the distributive property.

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 4 \cdot 9 + 4 (- x) - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.12.4

Multiply $4$ by $9$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 36 + 4 (- x) - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.12.5

Multiply $- 1$ by $4$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 6 x + 36 - 4 x - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.13

Subtract $4 x$ from $6 x$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 x + 36 - 72))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.14

Subtract $72$ from $36$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 2 x - 18 + 12 + 2 x - 36))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.15

Subtract $2 x$ from $x$ .

$f^{4} (x) = \frac{3 (3 (- x + 18 - 18 + 12 + 2 x - 36))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.16

Add $- x$ and $2 x$ .

$f^{4} (x) = \frac{3 (3 (x + 18 - 18 + 12 - 36))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.17

Subtract $18$ from $18$ .

$f^{4} (x) = \frac{3 (3 (x + 0 + 12 - 36))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.18

Add $x$ and $0$ .

$f^{4} (x) = \frac{3 (3 (x + 12 - 36))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.19

Subtract $36$ from $12$ .

$f^{4} (x) = \frac{3 \cdot (3 (x - 24))}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 4.84.8.20

Multiply $3$ by $3$ .

$f^{4} (x) = \frac{9 (x - 24)}{16 {(9 - x)}^{\frac{7}{2}}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $\frac{9 (x - 24)}{16 {(9 - x)}^{\frac{7}{2}}}$ .

$\frac{9 (x - 24)}{16 {(9 - x)}^{\frac{7}{2}}}$