Calculus Examples

$\frac{16 x - 5 x^{2}}{2 \sqrt{4 - x}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{16 x - 5 x^{2}}{2 {(4 - x)}^{\frac{1}{2}}}]$

Step 1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{16 x - 5 x^{2}}{2 {(4 - x)}^{\frac{1}{2}}}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{16 x - 5 x^{2}}{{(4 - x)}^{\frac{1}{2}}}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{16 x - 5 x^{2}}{{(4 - x)}^{\frac{1}{2}}}]$

Step 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 $f (x) = 16 x - 5 x^{2}$ and $g (x) = {(4 - x)}^{\frac{1}{2}}$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} \frac{d}{d x} [16 x - 5 x^{2}] - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{{({(4 - x)}^{\frac{1}{2}})}^{2}}$

Step 3

Multiply the exponents in ${({(4 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} \frac{d}{d x} [16 x - 5 x^{2}] - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{{(4 - x)}^{\frac{1}{2} \cdot 2}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} \frac{d}{d x} [16 x - 5 x^{2}] - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{{(4 - x)}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Rewrite the expression.

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} \frac{d}{d x} [16 x - 5 x^{2}] - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{{(4 - x)}^{1}}$

Step 4

Simplify.

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} \frac{d}{d x} [16 x - 5 x^{2}] - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $16 x - 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [16 x] + \frac{d}{d x} [- 5 x^{2}]$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (\frac{d}{d x} [16 x] + \frac{d}{d x} [- 5 x^{2}]) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5.2

Since $16$ is constant with respect to $x$ , the derivative of $16 x$ with respect to $x$ is $16 \frac{d}{d x} [x]$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 \frac{d}{d x} [x] + \frac{d}{d x} [- 5 x^{2}]) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5.3

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 \cdot 1 + \frac{d}{d x} [- 5 x^{2}]) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5.4

Multiply $16$ by $1$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 + \frac{d}{d x} [- 5 x^{2}]) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 5 \frac{d}{d x} [x^{2}]) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5.6

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 5 (2 x)) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 5.7

Multiply $2$ by $- 5$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}{4 - x}$

Step 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 $g (x) = 4 - x$ .

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Step 6.1

To apply the Chain Rule, set $u$ as $4 - x$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 - x])}{4 - x}$

Step 6.2

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x])}{4 - x}$

Step 6.3

Replace all occurrences of $u$ with $4 - x$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x])}{4 - x}$

Step 7

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 8

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 9

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{\frac{- 1}{2}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2} {(4 - x)}^{- \frac{1}{2}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 11.2

Combine $\frac{1}{2}$ and ${(4 - x)}^{- \frac{1}{2}}$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{{(4 - x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [4 - x])}{4 - x}$

Step 11.3

Move ${(4 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} \frac{d}{d x} [4 - x])}{4 - x}$

Step 12

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [4] + \frac{d}{d x} [- x]))}{4 - x}$

Step 13

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x]))}{4 - x}$

Step 14

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} \frac{d}{d x} [- x])}{4 - x}$

Step 15

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) - (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (- \frac{d}{d x} [x]))}{4 - x}$

Step 16

Multiply.

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Step 16.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + 1 (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [x]))}{4 - x}$

Step 16.2

Multiply $16 x - 5 x^{2}$ by $1$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + (16 x - 5 x^{2}) (\frac{1}{2 {(4 - x)}^{\frac{1}{2}}} (\frac{d}{d x} [x]))}{4 - x}$

Step 17

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

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}} \cdot 1}{4 - x}$

Step 18

Combine fractions.

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Step 18.1

Multiply $16 x - 5 x^{2}$ by $1$ .

$\frac{1}{2} \cdot \frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{4 - x}$

Step 18.2

Multiply $\frac{1}{2}$ by $\frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{4 - x}$ .

$\frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{2 (4 - x)}$

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

$\frac{{(4 - x)}^{\frac{1}{2}} (16 - 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2

Simplify the numerator.

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Step 19.2.1

Apply the distributive property.

$\frac{{(4 - x)}^{\frac{1}{2}} \cdot 16 + {(4 - x)}^{\frac{1}{2}} (- 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.2

Move $16$ to the left of ${(4 - x)}^{\frac{1}{2}}$ .

$\frac{16 \cdot {(4 - x)}^{\frac{1}{2}} + {(4 - x)}^{\frac{1}{2}} (- 10 x) + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.3

Rewrite using the commutative property of multiplication.

$\frac{16 \cdot {(4 - x)}^{\frac{1}{2}} - 10 {(4 - x)}^{\frac{1}{2}} x + (16 x - 5 x^{2}) \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.4

Multiply $16 x - 5 x^{2}$ by $\frac{1}{2 {(4 - x)}^{\frac{1}{2}}}$ .

$\frac{16 {(4 - x)}^{\frac{1}{2}} - 10 {(4 - x)}^{\frac{1}{2}} x + \frac{16 x - 5 x^{2}}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.5

Factor $x$ out of $16 x - 5 x^{2}$ .

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Step 19.2.5.1

Factor $x$ out of $16 x$ .

$\frac{16 {(4 - x)}^{\frac{1}{2}} - 10 {(4 - x)}^{\frac{1}{2}} x + \frac{x \cdot 16 - 5 x^{2}}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.5.2

Factor $x$ out of $- 5 x^{2}$ .

$\frac{16 {(4 - x)}^{\frac{1}{2}} - 10 {(4 - x)}^{\frac{1}{2}} x + \frac{x \cdot 16 + x (- 5 x)}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.5.3

Factor $x$ out of $x \cdot 16 + x (- 5 x)$ .

$\frac{16 {(4 - x)}^{\frac{1}{2}} - 10 {(4 - x)}^{\frac{1}{2}} x + \frac{x (16 - 5 x)}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.6

To write $16 {(4 - x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(4 - x)}^{\frac{1}{2}}}{2 {(4 - x)}^{\frac{1}{2}}}$ .

$\frac{- 10 {(4 - x)}^{\frac{1}{2}} x + 16 {(4 - x)}^{\frac{1}{2}} \cdot \frac{2 {(4 - x)}^{\frac{1}{2}}}{2 {(4 - x)}^{\frac{1}{2}}} + \frac{x (16 - 5 x)}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.7

Combine $16 {(4 - x)}^{\frac{1}{2}}$ and $\frac{2 {(4 - x)}^{\frac{1}{2}}}{2 {(4 - x)}^{\frac{1}{2}}}$ .

$\frac{- 10 {(4 - x)}^{\frac{1}{2}} x + \frac{16 {(4 - x)}^{\frac{1}{2}} (2 {(4 - x)}^{\frac{1}{2}})}{2 {(4 - x)}^{\frac{1}{2}}} + \frac{x (16 - 5 x)}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.8

Combine the numerators over the common denominator.

$\frac{- 10 {(4 - x)}^{\frac{1}{2}} x + \frac{16 {(4 - x)}^{\frac{1}{2}} (2 {(4 - x)}^{\frac{1}{2}}) + x (16 - 5 x)}{2 {(4 - x)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.9

Reorder $4$ and $- x$ .

$\frac{- 10 x {(- x + 4)}^{\frac{1}{2}} + \frac{x (- 5 x + 16) + 16 \cdot 2 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.10

To write $- 10 x {(- x + 4)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}$ .

$\frac{- 10 x {(- x + 4)}^{\frac{1}{2}} \cdot \frac{2 {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}} + \frac{x (- 5 x + 16) + 16 \cdot 2 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.11

Combine $- 10 x {(- x + 4)}^{\frac{1}{2}}$ and $\frac{2 {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}$ .

$\frac{\frac{- 10 x {(- x + 4)}^{\frac{1}{2}} (2 {(- x + 4)}^{\frac{1}{2}})}{2 {(- x + 4)}^{\frac{1}{2}}} + \frac{x (- 5 x + 16) + 16 \cdot 2 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.12

Combine the numerators over the common denominator.

$\frac{\frac{- 10 x {(- x + 4)}^{\frac{1}{2}} (2 {(- x + 4)}^{\frac{1}{2}}) + x (- 5 x + 16) + 16 \cdot 2 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.13

Reorder terms.

$\frac{\frac{- 10 \cdot 2 x {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}} + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14

Rewrite $\frac{- 10 \cdot 2 x {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}} + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}$ in a factored form.

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Step 19.2.14.1

Multiply ${(- x + 4)}^{\frac{1}{2}}$ by ${(- x + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 19.2.14.1.1

Move ${(- x + 4)}^{\frac{1}{2}}$ .

$\frac{\frac{- 10 \cdot 2 x ({(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}) + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.1.2

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

$\frac{\frac{- 10 \cdot 2 x {(- x + 4)}^{\frac{1}{2} + \frac{1}{2}} + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.1.3

Combine the numerators over the common denominator.

$\frac{\frac{- 10 \cdot 2 x {(- x + 4)}^{\frac{1 + 1}{2}} + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.1.4

Add $1$ and $1$ .

$\frac{\frac{- 10 \cdot 2 x {(- x + 4)}^{\frac{2}{2}} + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.1.5

Divide $2$ by $2$ .

$\frac{\frac{- 10 \cdot 2 x {(- x + 4)}^{1} + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.2

Simplify $- 10 \cdot 2 x {(- x + 4)}^{1}$ .

$\frac{\frac{- 10 \cdot 2 x (- x + 4) + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.3

Multiply $- 10$ by $2$ .

$\frac{\frac{- 20 x (- x + 4) + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.4

Apply the distributive property.

$\frac{\frac{- 20 x (- x) - 20 x \cdot 4 + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.5

Rewrite using the commutative property of multiplication.

$\frac{\frac{- 20 \cdot - 1 x \cdot x - 20 x \cdot 4 + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.6

Multiply $4$ by $- 20$ .

$\frac{\frac{- 20 \cdot - 1 x \cdot x - 80 x + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.7

Simplify each term.

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Step 19.2.14.7.1

Multiply $x$ by $x$ by adding the exponents.

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Step 19.2.14.7.1.1

Move $x$ .

$\frac{\frac{- 20 \cdot - 1 (x \cdot x) - 80 x + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.7.1.2

Multiply $x$ by $x$ .

$\frac{\frac{- 20 \cdot - 1 x^{2} - 80 x + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.7.2

Multiply $- 20$ by $- 1$ .

$\frac{\frac{20 x^{2} - 80 x + x (- 5 x + 16) + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.8

Apply the distributive property.

$\frac{\frac{20 x^{2} - 80 x + x (- 5 x) + x \cdot 16 + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.9

Rewrite using the commutative property of multiplication.

$\frac{\frac{20 x^{2} - 80 x - 5 x \cdot x + x \cdot 16 + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.10

Move $16$ to the left of $x$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x \cdot x + 16 \cdot x + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.11

Multiply $x$ by $x$ by adding the exponents.

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Step 19.2.14.11.1

Move $x$ .

$\frac{\frac{20 x^{2} - 80 x - 5 (x \cdot x) + 16 \cdot x + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.11.2

Multiply $x$ by $x$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 \cdot x + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.12

Multiply ${(- x + 4)}^{\frac{1}{2}}$ by ${(- x + 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 19.2.14.12.1

Move ${(- x + 4)}^{\frac{1}{2}}$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 ({(- x + 4)}^{\frac{1}{2}} {(- x + 4)}^{\frac{1}{2}})}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.12.2

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

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 {(- x + 4)}^{\frac{1}{2} + \frac{1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.12.3

Combine the numerators over the common denominator.

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 {(- x + 4)}^{\frac{1 + 1}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.12.4

Add $1$ and $1$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 {(- x + 4)}^{\frac{2}{2}}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.12.5

Divide $2$ by $2$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 {(- x + 4)}^{1}}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.13

Simplify $2 \cdot 16 {(- x + 4)}^{1}$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 2 \cdot 16 (- x + 4)}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.14

Multiply $2$ by $16$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 32 (- x + 4)}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.15

Apply the distributive property.

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x + 32 (- x) + 32 \cdot 4}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.16

Multiply $- 1$ by $32$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x - 32 x + 32 \cdot 4}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.17

Multiply $32$ by $4$ .

$\frac{\frac{20 x^{2} - 80 x - 5 x^{2} + 16 x - 32 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.18

Subtract $5 x^{2}$ from $20 x^{2}$ .

$\frac{\frac{15 x^{2} - 80 x + 16 x - 32 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.19

Add $- 80 x$ and $16 x$ .

$\frac{\frac{15 x^{2} - 64 x - 32 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.2.14.20

Subtract $32 x$ from $- 64 x$ .

$\frac{\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{2 \cdot 4 + 2 (- x)}$

Step 19.3

Combine terms.

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Step 19.3.1

Multiply $2$ by $4$ .

$\frac{\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{8 + 2 (- x)}$

Step 19.3.2

Multiply $- 1$ by $2$ .

$\frac{\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{8 - 2 x}$

Step 19.3.3

Rewrite $\frac{\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}}{8 - 2 x}$ as a product.

$\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}} \cdot \frac{1}{8 - 2 x}$

Step 19.3.4

Multiply $\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}}}$ by $\frac{1}{8 - 2 x}$ .

$\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}} (8 - 2 x)}$

Step 19.4

Simplify the denominator.

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Step 19.4.1

Factor $2$ out of $8 - 2 x$ .

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Step 19.4.1.1

Factor $2$ out of $8$ .

$\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}} (2 (4) - 2 x)}$

Step 19.4.1.2

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

$\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}} (2 (4) + 2 (- x))}$

Step 19.4.1.3

Factor $2$ out of $2 (4) + 2 (- x)$ .

$\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}} (2 (4 - x))}$

$\frac{15 x^{2} - 96 x + 128}{2 {(- x + 4)}^{\frac{1}{2}} \cdot 2 (4 - x)}$

Step 19.4.2

Combine exponents.

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Step 19.4.2.1

Multiply $2$ by $2$ .

$\frac{15 x^{2} - 96 x + 128}{4 {(- x + 4)}^{\frac{1}{2}} (4 - x)}$

Step 19.4.2.2

Reorder terms.

$\frac{15 x^{2} - 96 x + 128}{4 {(- x + 4)}^{\frac{1}{2}} (- x + 4)}$

Step 19.4.2.3

Raise $- x + 4$ to the power of $1$ .

$\frac{15 x^{2} - 96 x + 128}{4 ({(- x + 4)}^{1} {(- x + 4)}^{\frac{1}{2}})}$

Step 19.4.2.4

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

$\frac{15 x^{2} - 96 x + 128}{4 {(- x + 4)}^{1 + \frac{1}{2}}}$

Step 19.4.2.5

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

$\frac{15 x^{2} - 96 x + 128}{4 {(- x + 4)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 19.4.2.6

Combine the numerators over the common denominator.

$\frac{15 x^{2} - 96 x + 128}{4 {(- x + 4)}^{\frac{2 + 1}{2}}}$

Step 19.4.2.7

Add $2$ and $1$ .

$\frac{15 x^{2} - 96 x + 128}{4 {(- x + 4)}^{\frac{3}{2}}}$