Calculus Examples

$f (x) = \frac{36 - 2 x^{2}}{\sqrt{36 - x^{2}}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{36 - x^{2}}$ as ${(36 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{36 - 2 x^{2}}{{(36 - x^{2})}^{\frac{1}{2}}}]$

Step 1.1.2

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 $f (x) = 36 - 2 x^{2}$ and $g (x) = {(36 - x^{2})}^{\frac{1}{2}}$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [36 - 2 x^{2}] - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{{({(36 - x^{2})}^{\frac{1}{2}})}^{2}}$

Step 1.1.3

Multiply the exponents in ${({(36 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.1.3.1

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [36 - 2 x^{2}] - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{{(36 - x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 1.1.3.2

Cancel the common factor of $2$ .

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Step 1.1.3.2.1

Cancel the common factor.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [36 - 2 x^{2}] - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{{(36 - x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 1.1.3.2.2

Rewrite the expression.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [36 - 2 x^{2}] - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{{(36 - x^{2})}^{1}}$

Step 1.1.4

Simplify.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [36 - 2 x^{2}] - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.5

Differentiate.

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Step 1.1.5.1

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (\frac{d}{d x} [36] + \frac{d}{d x} [- 2 x^{2}]) - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.5.2

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (0 + \frac{d}{d x} [- 2 x^{2}]) - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.5.3

Add $0$ and $\frac{d}{d x} [- 2 x^{2}]$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [- 2 x^{2}] - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.5.4

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 2 \frac{d}{d x} [x^{2}]) - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.5.5

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 2 (2 x)) - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.5.6

Multiply $2$ by $- 2$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) \frac{d}{d x} [{(36 - x^{2})}^{\frac{1}{2}}]}{36 - x^{2}}$

Step 1.1.6

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 $g (x) = 36 - x^{2}$ .

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Step 1.1.6.1

To apply the Chain Rule, set $u$ as $36 - x^{2}$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.6.2

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.6.3

Replace all occurrences of $u$ with $36 - x^{2}$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.7

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.8

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.9

Combine the numerators over the common denominator.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.10

Simplify the numerator.

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Step 1.1.10.1

Multiply $- 1$ by $2$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.10.2

Subtract $2$ from $1$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.11

Combine fractions.

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Step 1.1.11.1

Move the negative in front of the fraction.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2} {(36 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.11.2

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{{(36 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.11.3

Move ${(36 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [36 - x^{2}])}{36 - x^{2}}$

Step 1.1.12

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [36] + \frac{d}{d x} [- x^{2}]))}{36 - x^{2}}$

Step 1.1.13

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]))}{36 - x^{2}}$

Step 1.1.14

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}])}{36 - x^{2}}$

Step 1.1.15

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) - (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]))}{36 - x^{2}}$

Step 1.1.16

Multiply.

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Step 1.1.16.1

Multiply $- 1$ by $- 1$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + 1 (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}]))}{36 - x^{2}}$

Step 1.1.16.2

Multiply $36 - 2 x^{2}$ by $1$ .

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) (\frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}]))}{36 - x^{2}}$

Step 1.1.17

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) \frac{1}{2 {(36 - x^{2})}^{\frac{1}{2}}} (2 x)}{36 - x^{2}}$

Step 1.1.18

Simplify terms.

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Step 1.1.18.1

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) \frac{2}{2 {(36 - x^{2})}^{\frac{1}{2}}} x}{36 - x^{2}}$

Step 1.1.18.2

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) \frac{x \cdot 2}{2 {(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.18.3

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

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) \frac{2 \cdot x}{2 {(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.18.4

Cancel the common factor.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) \frac{2 x}{2 {(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.18.5

Rewrite the expression.

$\frac{{(36 - x^{2})}^{\frac{1}{2}} (- 4 x) + (36 - 2 x^{2}) \frac{x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19

Simplify.

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Step 1.1.19.1

Simplify the numerator.

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Step 1.1.19.1.1

Rewrite using the commutative property of multiplication.

$\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x + (36 - 2 x^{2}) \frac{x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.2

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

$\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x + \frac{(36 - 2 x^{2}) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.3

Factor $2$ out of $36 - 2 x^{2}$ .

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Step 1.1.19.1.3.1

Factor $2$ out of $36$ .

$\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x + \frac{(2 (18) - 2 x^{2}) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.3.2

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

$\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x + \frac{(2 (18) + 2 (- x^{2})) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.3.3

Factor $2$ out of $2 (18) + 2 (- x^{2})$ .

$\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x + \frac{2 (18 - x^{2}) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.4

To write $- 4 {(36 - x^{2})}^{\frac{1}{2}} x$ as a fraction with a common denominator, multiply by $\frac{{(36 - x^{2})}^{\frac{1}{2}}}{{(36 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x \cdot \frac{{(36 - x^{2})}^{\frac{1}{2}}}{{(36 - x^{2})}^{\frac{1}{2}}} + \frac{2 (18 - x^{2}) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.5

Combine $- 4 {(36 - x^{2})}^{\frac{1}{2}} x$ and $\frac{{(36 - x^{2})}^{\frac{1}{2}}}{{(36 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x {(36 - x^{2})}^{\frac{1}{2}}}{{(36 - x^{2})}^{\frac{1}{2}}} + \frac{2 (18 - x^{2}) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.6

Combine the numerators over the common denominator.

$\frac{\frac{- 4 {(36 - x^{2})}^{\frac{1}{2}} x {(36 - x^{2})}^{\frac{1}{2}} + 2 (18 - x^{2}) x}{{(36 - x^{2})}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.7

Reorder terms.

$\frac{\frac{- 4 x {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}} + 2 x (18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8

Rewrite $\frac{- 4 x {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}} + 2 x (18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}$ in a factored form.

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Step 1.1.19.1.8.1

Factor $2 x$ out of $- 4 x {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}} + 2 x (18 - x^{2})$ .

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Step 1.1.19.1.8.1.1

Factor $2 x$ out of $- 4 x {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{2 x (- 2 {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}}) + 2 x (18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.1.2

Factor $2 x$ out of $2 x (- 2 {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}}) + 2 x (18 - x^{2})$ .

$\frac{\frac{2 x (- 2 {(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}} + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.2

Combine exponents.

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Step 1.1.19.1.8.2.1

Multiply ${(36 - x^{2})}^{\frac{1}{2}}$ by ${(36 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.1.19.1.8.2.1.1

Move ${(36 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{2 x (- 2 ({(36 - x^{2})}^{\frac{1}{2}} {(36 - x^{2})}^{\frac{1}{2}}) + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.2.1.2

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

$\frac{\frac{2 x (- 2 {(36 - x^{2})}^{\frac{1}{2} + \frac{1}{2}} + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.2.1.3

Combine the numerators over the common denominator.

$\frac{\frac{2 x (- 2 {(36 - x^{2})}^{\frac{1 + 1}{2}} + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.2.1.4

Add $1$ and $1$ .

$\frac{\frac{2 x (- 2 {(36 - x^{2})}^{\frac{2}{2}} + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.2.1.5

Divide $2$ by $2$ .

$\frac{\frac{2 x (- 2 {(36 - x^{2})}^{1} + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.8.2.2

Simplify $- 2 {(36 - x^{2})}^{1}$ .

$\frac{\frac{2 x (- 2 (36 - x^{2}) + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.9

Simplify the numerator.

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Step 1.1.19.1.9.1

Apply the distributive property.

$\frac{\frac{2 x (- 2 \cdot 36 - 2 (- x^{2}) + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.9.2

Multiply $- 2$ by $36$ .

$\frac{\frac{2 x (- 72 - 2 (- x^{2}) + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.9.3

Multiply $- 1$ by $- 2$ .

$\frac{\frac{2 x (- 72 + 2 x^{2} + 18 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.9.4

Add $- 72$ and $18$ .

$\frac{\frac{2 x (2 x^{2} - 54 - x^{2})}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.1.9.5

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

$\frac{\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$

Step 1.1.19.2

Combine terms.

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Step 1.1.19.2.1

Rewrite $\frac{\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}}}}{36 - x^{2}}$ as a product.

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}}} \cdot \frac{1}{36 - x^{2}}$

Step 1.1.19.2.2

Multiply $\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}}}$ by $\frac{1}{36 - x^{2}}$ .

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}} (36 - x^{2})}$

Step 1.1.19.2.3

Reorder terms.

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}} (- x^{2} + 36)}$

Step 1.1.19.2.4

Multiply ${(- x^{2} + 36)}^{\frac{1}{2}}$ by $- x^{2} + 36$ by adding the exponents.

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Step 1.1.19.2.4.1

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

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Step 1.1.19.2.4.1.1

Raise $- x^{2} + 36$ to the power of $1$ .

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2}} {(- x^{2} + 36)}^{1}}$

Step 1.1.19.2.4.1.2

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

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2} + 1}}$

Step 1.1.19.2.4.2

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

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 1.1.19.2.4.3

Combine the numerators over the common denominator.

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{1 + 2}{2}}}$

Step 1.1.19.2.4.4

Add $1$ and $2$ .

$f' (x) = \frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{3}{2}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{3}{2}}}$ .

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{3}{2}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{3}{2}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2 x (x^{2} - 54)}{{(- x^{2} + 36)}^{\frac{3}{2}}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 x (x^{2} - 54) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} - 54 = 0$

Step 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

Set $x^{2} - 54$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x^{2} - 54$ equal to $0$ .

$x^{2} - 54 = 0$

Step 2.3.3.2

Solve $x^{2} - 54 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $54$ to both sides of the equation.

$x^{2} = 54$

Step 2.3.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{54}$

Step 2.3.3.2.3

Simplify $\pm \sqrt{54}$ .

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Step 2.3.3.2.3.1

Rewrite $54$ as $3^{2} \cdot 6$ .

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Step 2.3.3.2.3.1.1

Factor $9$ out of $54$ .

$x = \pm \sqrt{9 (6)}$

Step 2.3.3.2.3.1.2

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2} \cdot 6}$

Step 2.3.3.2.3.2

Pull terms out from under the radical.

$x = \pm 3 \sqrt{6}$

Step 2.3.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 \sqrt{6}$

Step 2.3.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 \sqrt{6}$

Step 2.3.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 \sqrt{6}, - 3 \sqrt{6}$

Step 2.3.4

The final solution is all the values that make $2 x (x^{2} - 54) = 0$ true.

$x = 0, 3 \sqrt{6}, - 3 \sqrt{6}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{2 x (x^{2} - 54)}{\sqrt{{(- x^{2} + 36)}^{3}}}$

Step 3.2

Set the denominator in $\frac{2 x (x^{2} - 54)}{\sqrt{{(- x^{2} + 36)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{{(- x^{2} + 36)}^{3}} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{{(- x^{2} + 36)}^{3}}}^{2} = 0^{2}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(- x^{2} + 36)}^{3}}$ as ${(- x^{2} + 36)}^{\frac{3}{2}}$ .

${({(- x^{2} + 36)}^{\frac{3}{2}})}^{2} = 0^{2}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Multiply the exponents in ${({(- x^{2} + 36)}^{\frac{3}{2}})}^{2}$ .

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Step 3.3.2.2.1.1

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

${(- x^{2} + 36)}^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1.2.1

Cancel the common factor.

${(- x^{2} + 36)}^{\frac{3}{2} \cdot 2} = 0^{2}$

Step 3.3.2.2.1.2.2

Rewrite the expression.

${(- x^{2} + 36)}^{3} = 0^{2}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

${(- x^{2} + 36)}^{3} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Factor the left side of the equation.

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Step 3.3.3.1.1

Factor $- 1$ out of $- x^{2} + 36$ .

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Step 3.3.3.1.1.1

Factor $- 1$ out of $- x^{2}$ .

${(- (x^{2}) + 36)}^{3} = 0$

Step 3.3.3.1.1.2

Rewrite $36$ as $- 1 (- 36)$ .

${(- (x^{2}) - 1 \cdot - 36)}^{3} = 0$

Step 3.3.3.1.1.3

Factor $- 1$ out of $- (x^{2}) - 1 (- 36)$ .

${(- (x^{2} - 36))}^{3} = 0$

Step 3.3.3.1.2

Rewrite $36$ as $6^{2}$ .

${(- (x^{2} - 6^{2}))}^{3} = 0$

Step 3.3.3.1.3

Factor.

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Step 3.3.3.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

${(- ((x + 6) (x - 6)))}^{3} = 0$

Step 3.3.3.1.3.2

Remove unnecessary parentheses.

${(- (x + 6) (x - 6))}^{3} = 0$

Step 3.3.3.1.4

Apply the product rule to $- (x + 6) (x - 6)$ .

${(- (x + 6))}^{3} {(x - 6)}^{3} = 0$

Step 3.3.3.1.5

Apply the product rule to $- (x + 6)$ .

${(- 1)}^{3} {(x + 6)}^{3} {(x - 6)}^{3} = 0$

Step 3.3.3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 6)}^{3} = 0$

${(x - 6)}^{3} = 0$

Step 3.3.3.3

Set ${(x + 6)}^{3}$ equal to $0$ and solve for $x$ .

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Step 3.3.3.3.1

Set ${(x + 6)}^{3}$ equal to $0$ .

${(x + 6)}^{3} = 0$

Step 3.3.3.3.2

Solve ${(x + 6)}^{3} = 0$ for $x$ .

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Step 3.3.3.3.2.1

Set the $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 3.3.3.3.2.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 3.3.3.4

Set ${(x - 6)}^{3}$ equal to $0$ and solve for $x$ .

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Step 3.3.3.4.1

Set ${(x - 6)}^{3}$ equal to $0$ .

${(x - 6)}^{3} = 0$

Step 3.3.3.4.2

Solve ${(x - 6)}^{3} = 0$ for $x$ .

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Step 3.3.3.4.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.3.3.4.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.3.3.5

The final solution is all the values that make ${(- 1)}^{3} {(x + 6)}^{3} {(x - 6)}^{3} = 0$ true.

$x = - 6, 6$

Step 3.4

Set the radicand in $\sqrt{{(- x^{2} + 36)}^{3}}$ less than $0$ to find where the expression is undefined.

${(- x^{2} + 36)}^{3} < 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{{(- x^{2} + 36)}^{3}} < \sqrt[3]{0}$

Step 3.5.2

Simplify the equation.

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Step 3.5.2.1

Simplify the left side.

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Step 3.5.2.1.1

Pull terms out from under the radical.

$- x^{2} + 36 < \sqrt[3]{0}$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$- x^{2} + 36 < \sqrt[3]{0^{3}}$

Step 3.5.2.2.1.2

Pull terms out from under the radical.

$- x^{2} + 36 < 0$

Step 3.5.3

Subtract $36$ from both sides of the inequality.

$- x^{2} < - 36$

Step 3.5.4

Divide each term in $- x^{2} < - 36$ by $- 1$ and simplify.

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Step 3.5.4.1

Divide each term in $- x^{2} < - 36$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 36}{- 1}$

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 36}{- 1}$

Step 3.5.4.2.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{- 36}{- 1}$

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Divide $- 36$ by $- 1$ .

$x^{2} > 36$

Step 3.5.5

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} > \sqrt{36}$

Step 3.5.6

Simplify the equation.

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Step 3.5.6.1

Simplify the left side.

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Step 3.5.6.1.1

Pull terms out from under the radical.

$| x | > \sqrt{36}$

Step 3.5.6.2

Simplify the right side.

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Step 3.5.6.2.1

Simplify $\sqrt{36}$ .

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Step 3.5.6.2.1.1

Rewrite $36$ as $6^{2}$ .

$| x | > \sqrt{6^{2}}$

Step 3.5.6.2.1.2

Pull terms out from under the radical.

$| x | > | 6 |$

Step 3.5.6.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$| x | > 6$

Step 3.5.7

Write $| x | > 6$ as a piecewise.

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Step 3.5.7.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 3.5.7.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 6$

Step 3.5.7.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 3.5.7.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 6$

Step 3.5.7.5

Write as a piecewise.

${\begin{cases} x > 6 & x \geq 0 \\ - x > 6 & x < 0 \end{cases}$

Step 3.5.8

Find the intersection of $x > 6$ and $x \geq 0$ .

$x > 6$

Step 3.5.9

Divide each term in $- x > 6$ by $- 1$ and simplify.

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Step 3.5.9.1

Divide each term in $- x > 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{6}{- 1}$

Step 3.5.9.2

Simplify the left side.

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Step 3.5.9.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{6}{- 1}$

Step 3.5.9.2.2

Divide $x$ by $1$ .

$x < \frac{6}{- 1}$

Step 3.5.9.3

Simplify the right side.

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Step 3.5.9.3.1

Divide $6$ by $- 1$ .

$x < - 6$

Step 3.5.10

Find the union of the solutions.

$x < - 6$ or $x > 6$

Step 3.6

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x \leq - 6, x \geq 6$

$(- \infty, - 6] \cup [6, \infty)$

$x \leq - 6, x \geq 6$

$(- \infty, - 6] \cup [6, \infty)$

Step 4

Evaluate $\frac{36 - 2 x^{2}}{\sqrt{36 - x^{2}}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{36 - 2 {(0)}^{2}}{\sqrt{36 - {(0)}^{2}}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{36 - 2 \cdot 0}{\sqrt{36 - 0^{2}}}$

Step 4.1.2.1.2

Multiply $- 2$ by $0$ .

$\frac{36 + 0}{\sqrt{36 - 0^{2}}}$

Step 4.1.2.1.3

Add $36$ and $0$ .

$\frac{36}{\sqrt{36 - 0^{2}}}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{36}{\sqrt{36 - 0}}$

Step 4.1.2.2.2

Multiply $- 1$ by $0$ .

$\frac{36}{\sqrt{36 + 0}}$

Step 4.1.2.2.3

Add $36$ and $0$ .

$\frac{36}{\sqrt{36}}$

Step 4.1.2.2.4

Rewrite $36$ as $6^{2}$ .

$\frac{36}{\sqrt{6^{2}}}$

Step 4.1.2.2.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{36}{6}$

Step 4.1.2.3

Divide $36$ by $6$ .

$6$

Step 4.2

Evaluate at $x = 3 \sqrt{6}$ .

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Step 4.2.1

Substitute $3 \sqrt{6}$ for $x$ .

$\frac{36 - 2 {(3 \sqrt{6})}^{2}}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Apply the product rule to $3 \sqrt{6}$ .

$\frac{36 - 2 (3^{2} {\sqrt{6}}^{2})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.2

Raise $3$ to the power of $2$ .

$\frac{36 - 2 (9 {\sqrt{6}}^{2})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.2.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{36 - 2 (9 {(6^{\frac{1}{2}})}^{2})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.3.2

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

$\frac{36 - 2 (9 \cdot 6^{\frac{1}{2} \cdot 2})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.3.3

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

$\frac{36 - 2 (9 \cdot 6^{\frac{2}{2}})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.4.1

Cancel the common factor.

$\frac{36 - 2 (9 \cdot 6^{\frac{2}{2}})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.3.4.2

Rewrite the expression.

$\frac{36 - 2 (9 \cdot 6^{1})}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.3.5

Evaluate the exponent.

$\frac{36 - 2 (9 \cdot 6)}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.4

Multiply $- 2 (9 \cdot 6)$ .

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Step 4.2.2.1.4.1

Multiply $9$ by $6$ .

$\frac{36 - 2 \cdot 54}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.4.2

Multiply $- 2$ by $54$ .

$\frac{36 - 108}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.1.5

Subtract $108$ from $36$ .

$\frac{- 72}{\sqrt{36 - {(3 \sqrt{6})}^{2}}}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Apply the product rule to $3 \sqrt{6}$ .

$\frac{- 72}{\sqrt{36 - (3^{2} {\sqrt{6}}^{2})}}$

Step 4.2.2.2.2

Raise $3$ to the power of $2$ .

$\frac{- 72}{\sqrt{36 - (9 {\sqrt{6}}^{2})}}$

Step 4.2.2.2.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.2.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{- 72}{\sqrt{36 - (9 {(6^{\frac{1}{2}})}^{2})}}$

Step 4.2.2.2.3.2

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

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{\frac{1}{2} \cdot 2})}}$

Step 4.2.2.2.3.3

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

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{\frac{2}{2}})}}$

Step 4.2.2.2.3.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.3.4.1

Cancel the common factor.

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{\frac{2}{2}})}}$

Step 4.2.2.2.3.4.2

Rewrite the expression.

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{1})}}$

Step 4.2.2.2.3.5

Evaluate the exponent.

$\frac{- 72}{\sqrt{36 - (9 \cdot 6)}}$

Step 4.2.2.2.4

Multiply $- (9 \cdot 6)$ .

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Step 4.2.2.2.4.1

Multiply $9$ by $6$ .

$\frac{- 72}{\sqrt{36 - 1 \cdot 54}}$

Step 4.2.2.2.4.2

Multiply $- 1$ by $54$ .

$\frac{- 72}{\sqrt{36 - 54}}$

Step 4.2.2.2.5

Subtract $54$ from $36$ .

$\frac{- 72}{\sqrt{- 18}}$

Step 4.2.2.2.6

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- 72}{\sqrt{- 1 (18)}}$

Step 4.2.2.2.7

Rewrite $\sqrt{- 1 (18)}$ as $\sqrt{- 1} \cdot \sqrt{18}$ .

$\frac{- 72}{\sqrt{- 1} \cdot \sqrt{18}}$

Step 4.2.2.2.8

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{- 72}{i \cdot \sqrt{18}}$

Step 4.2.2.2.9

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.2.2.2.9.1

Factor $9$ out of $18$ .

$\frac{- 72}{i \cdot \sqrt{9 (2)}}$

Step 4.2.2.2.9.2

Rewrite $9$ as $3^{2}$ .

$\frac{- 72}{i \cdot \sqrt{3^{2} \cdot 2}}$

Step 4.2.2.2.10

Pull terms out from under the radical.

$\frac{- 72}{i \cdot (3 \sqrt{2})}$

Step 4.2.2.2.11

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

$\frac{- 72}{3 i \sqrt{2}}$

Step 4.2.2.3

Cancel the common factor of $- 72$ and $3$ .

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Step 4.2.2.3.1

Factor $3$ out of $- 72$ .

$\frac{3 \cdot - 24}{3 i \sqrt{2}}$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $3$ out of $3 i \sqrt{2}$ .

$\frac{3 \cdot - 24}{3 (i \sqrt{2})}$

Step 4.2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 24}{3 (i \sqrt{2})}$

Step 4.2.2.3.2.3

Rewrite the expression.

$\frac{- 24}{i \sqrt{2}}$

Step 4.2.2.4

Multiply the numerator and denominator of $\frac{- 24}{1.41421356 i}$ by the conjugate of $1.41421356 i$ to make the denominator real.

$\frac{- 24}{1.41421356 i} \cdot \frac{i}{i}$

Step 4.2.2.5

Multiply.

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Step 4.2.2.5.1

Combine.

$\frac{- 24 i}{1.41421356 i i}$

Step 4.2.2.5.2

Simplify the denominator.

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Step 4.2.2.5.2.1

Add parentheses.

$\frac{- 24 i}{1.41421356 (i i)}$

Step 4.2.2.5.2.2

Raise $i$ to the power of $1$ .

$\frac{- 24 i}{1.41421356 (i^{1} i)}$

Step 4.2.2.5.2.3

Raise $i$ to the power of $1$ .

$\frac{- 24 i}{1.41421356 (i^{1} i^{1})}$

Step 4.2.2.5.2.4

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

$\frac{- 24 i}{1.41421356 i^{1 + 1}}$

Step 4.2.2.5.2.5

Add $1$ and $1$ .

$\frac{- 24 i}{1.41421356 i^{2}}$

Step 4.2.2.5.2.6

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 24 i}{1.41421356 \cdot - 1}$

Step 4.2.2.6

Multiply $1.41421356$ by $- 1$ .

$\frac{- 24 i}{- 1.41421356}$

Step 4.2.2.7

Dividing two negative values results in a positive value.

$\frac{24 i}{1.41421356}$

Step 4.2.2.8

Factor $24$ out of $24 i$ .

$\frac{24 (i)}{1.41421356}$

Step 4.2.2.9

Factor $1.41421356$ out of $1.41421356$ .

$\frac{24 (i)}{1.41421356 (1)}$

Step 4.2.2.10

Separate fractions.

$\frac{24}{1.41421356} \cdot \frac{i}{1}$

Step 4.2.2.11

Divide $24$ by $1.41421356$ .

$16.97056274 \frac{i}{1}$

Step 4.2.2.12

Divide $i$ by $1$ .

$16.97056274 i$

Step 4.3

Evaluate at $x = - 3 \sqrt{6}$ .

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Step 4.3.1

Substitute $- 3 \sqrt{6}$ for $x$ .

$\frac{36 - 2 {(- 3 \sqrt{6})}^{2}}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the numerator.

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Step 4.3.2.1.1

Apply the product rule to $- 3 \sqrt{6}$ .

$\frac{36 - 2 ({(- 3)}^{2} {\sqrt{6}}^{2})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.2

Raise $- 3$ to the power of $2$ .

$\frac{36 - 2 (9 {\sqrt{6}}^{2})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.3.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{36 - 2 (9 {(6^{\frac{1}{2}})}^{2})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.3.2

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

$\frac{36 - 2 (9 \cdot 6^{\frac{1}{2} \cdot 2})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.3.3

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

$\frac{36 - 2 (9 \cdot 6^{\frac{2}{2}})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.3.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.3.4.1

Cancel the common factor.

$\frac{36 - 2 (9 \cdot 6^{\frac{2}{2}})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.3.4.2

Rewrite the expression.

$\frac{36 - 2 (9 \cdot 6^{1})}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.3.5

Evaluate the exponent.

$\frac{36 - 2 (9 \cdot 6)}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.4

Multiply $- 2 (9 \cdot 6)$ .

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Step 4.3.2.1.4.1

Multiply $9$ by $6$ .

$\frac{36 - 2 \cdot 54}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.4.2

Multiply $- 2$ by $54$ .

$\frac{36 - 108}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.1.5

Subtract $108$ from $36$ .

$\frac{- 72}{\sqrt{36 - {(- 3 \sqrt{6})}^{2}}}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

Apply the product rule to $- 3 \sqrt{6}$ .

$\frac{- 72}{\sqrt{36 - ({(- 3)}^{2} {\sqrt{6}}^{2})}}$

Step 4.3.2.2.2

Raise $- 3$ to the power of $2$ .

$\frac{- 72}{\sqrt{36 - (9 {\sqrt{6}}^{2})}}$

Step 4.3.2.2.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.3.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{- 72}{\sqrt{36 - (9 {(6^{\frac{1}{2}})}^{2})}}$

Step 4.3.2.2.3.2

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

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{\frac{1}{2} \cdot 2})}}$

Step 4.3.2.2.3.3

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

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{\frac{2}{2}})}}$

Step 4.3.2.2.3.4

Cancel the common factor of $2$ .

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Step 4.3.2.2.3.4.1

Cancel the common factor.

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{\frac{2}{2}})}}$

Step 4.3.2.2.3.4.2

Rewrite the expression.

$\frac{- 72}{\sqrt{36 - (9 \cdot 6^{1})}}$

Step 4.3.2.2.3.5

Evaluate the exponent.

$\frac{- 72}{\sqrt{36 - (9 \cdot 6)}}$

Step 4.3.2.2.4

Multiply $- (9 \cdot 6)$ .

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Step 4.3.2.2.4.1

Multiply $9$ by $6$ .

$\frac{- 72}{\sqrt{36 - 1 \cdot 54}}$

Step 4.3.2.2.4.2

Multiply $- 1$ by $54$ .

$\frac{- 72}{\sqrt{36 - 54}}$

Step 4.3.2.2.5

Subtract $54$ from $36$ .

$\frac{- 72}{\sqrt{- 18}}$

Step 4.3.2.2.6

Rewrite $- 18$ as $- 1 (18)$ .

$\frac{- 72}{\sqrt{- 1 (18)}}$

Step 4.3.2.2.7

Rewrite $\sqrt{- 1 (18)}$ as $\sqrt{- 1} \cdot \sqrt{18}$ .

$\frac{- 72}{\sqrt{- 1} \cdot \sqrt{18}}$

Step 4.3.2.2.8

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{- 72}{i \cdot \sqrt{18}}$

Step 4.3.2.2.9

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.3.2.2.9.1

Factor $9$ out of $18$ .

$\frac{- 72}{i \cdot \sqrt{9 (2)}}$

Step 4.3.2.2.9.2

Rewrite $9$ as $3^{2}$ .

$\frac{- 72}{i \cdot \sqrt{3^{2} \cdot 2}}$

Step 4.3.2.2.10

Pull terms out from under the radical.

$\frac{- 72}{i \cdot (3 \sqrt{2})}$

Step 4.3.2.2.11

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

$\frac{- 72}{3 i \sqrt{2}}$

Step 4.3.2.3

Cancel the common factor of $- 72$ and $3$ .

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Step 4.3.2.3.1

Factor $3$ out of $- 72$ .

$\frac{3 \cdot - 24}{3 i \sqrt{2}}$

Step 4.3.2.3.2

Cancel the common factors.

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Step 4.3.2.3.2.1

Factor $3$ out of $3 i \sqrt{2}$ .

$\frac{3 \cdot - 24}{3 (i \sqrt{2})}$

Step 4.3.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 24}{3 (i \sqrt{2})}$

Step 4.3.2.3.2.3

Rewrite the expression.

$\frac{- 24}{i \sqrt{2}}$

Step 4.3.2.4

Multiply the numerator and denominator of $\frac{- 24}{1.41421356 i}$ by the conjugate of $1.41421356 i$ to make the denominator real.

$\frac{- 24}{1.41421356 i} \cdot \frac{i}{i}$

Step 4.3.2.5

Multiply.

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Step 4.3.2.5.1

Combine.

$\frac{- 24 i}{1.41421356 i i}$

Step 4.3.2.5.2

Simplify the denominator.

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Step 4.3.2.5.2.1

Add parentheses.

$\frac{- 24 i}{1.41421356 (i i)}$

Step 4.3.2.5.2.2

Raise $i$ to the power of $1$ .

$\frac{- 24 i}{1.41421356 (i^{1} i)}$

Step 4.3.2.5.2.3

Raise $i$ to the power of $1$ .

$\frac{- 24 i}{1.41421356 (i^{1} i^{1})}$

Step 4.3.2.5.2.4

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

$\frac{- 24 i}{1.41421356 i^{1 + 1}}$

Step 4.3.2.5.2.5

Add $1$ and $1$ .

$\frac{- 24 i}{1.41421356 i^{2}}$

Step 4.3.2.5.2.6

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 24 i}{1.41421356 \cdot - 1}$

Step 4.3.2.6

Multiply $1.41421356$ by $- 1$ .

$\frac{- 24 i}{- 1.41421356}$

Step 4.3.2.7

Dividing two negative values results in a positive value.

$\frac{24 i}{1.41421356}$

Step 4.3.2.8

Factor $24$ out of $24 i$ .

$\frac{24 (i)}{1.41421356}$

Step 4.3.2.9

Factor $1.41421356$ out of $1.41421356$ .

$\frac{24 (i)}{1.41421356 (1)}$

Step 4.3.2.10

Separate fractions.

$\frac{24}{1.41421356} \cdot \frac{i}{1}$

Step 4.3.2.11

Divide $24$ by $1.41421356$ .

$16.97056274 \frac{i}{1}$

Step 4.3.2.12

Divide $i$ by $1$ .

$16.97056274 i$

Step 4.4

Evaluate at $x = - 6$ .

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Step 4.4.1

Substitute $- 6$ for $x$ .

$\frac{36 - 2 {(- 6)}^{2}}{\sqrt{36 - {(- 6)}^{2}}}$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Raise $- 6$ to the power of $2$ .

$\frac{36 - 2 {(- 6)}^{2}}{\sqrt{36 - 1 \cdot 36}}$

Step 4.4.2.2

Multiply $- 1$ by $36$ .

$\frac{36 - 2 {(- 6)}^{2}}{\sqrt{36 - 36}}$

Step 4.4.2.3

Subtract $36$ from $36$ .

$\frac{36 - 2 {(- 6)}^{2}}{\sqrt{0}}$

Step 4.4.2.4

Rewrite $0$ as $0^{2}$ .

$\frac{36 - 2 {(- 6)}^{2}}{\sqrt{0^{2}}}$

Step 4.4.2.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{36 - 2 {(- 6)}^{2}}{0}$

Step 4.4.2.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.5

Evaluate at $x = 6$ .

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Step 4.5.1

Substitute $6$ for $x$ .

$\frac{36 - 2 {(6)}^{2}}{\sqrt{36 - {(6)}^{2}}}$

Step 4.5.2

Simplify.

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Step 4.5.2.1

Raise $6$ to the power of $2$ .

$\frac{36 - 2 {(6)}^{2}}{\sqrt{36 - 1 \cdot 36}}$

Step 4.5.2.2

Multiply $- 1$ by $36$ .

$\frac{36 - 2 {(6)}^{2}}{\sqrt{36 - 36}}$

Step 4.5.2.3

Subtract $36$ from $36$ .

$\frac{36 - 2 {(6)}^{2}}{\sqrt{0}}$

Step 4.5.2.4

Rewrite $0$ as $0^{2}$ .

$\frac{36 - 2 {(6)}^{2}}{\sqrt{0^{2}}}$

Step 4.5.2.5

Pull terms out from under the radical, assuming positive real numbers.

$\frac{36 - 2 {(6)}^{2}}{0}$

Step 4.5.2.6

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.6

List all of the points.

$(0, 6)$

Step 5