Calculus Examples

$y = \frac{1}{2} \cdot (x \sqrt{64 - x^{2}} + 64 \arcsin (\frac{x}{8}))$

Step 1

Differentiate.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{64 - x^{2}}$ as ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{1}{2} (x {(64 - x^{2})}^{\frac{1}{2}} + 64 \arcsin (\frac{x}{8}))]$

Step 1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} (x {(64 - x^{2})}^{\frac{1}{2}} + 64 \arcsin (\frac{x}{8}))$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}} + 64 \arcsin (\frac{x}{8})]$ .

$\frac{1}{2} \frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}} + 64 \arcsin (\frac{x}{8})]$

Step 1.3

By the Sum Rule, the derivative of $x {(64 - x^{2})}^{\frac{1}{2}} + 64 \arcsin (\frac{x}{8})$ with respect to $x$ is $\frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}}] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})]$ .

$\frac{1}{2} (\frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}}] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 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 $g (x) = {(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{1}{2} (x \frac{d}{d x} [{(64 - x^{2})}^{\frac{1}{2}}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 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 $g (x) = 64 - x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $64 - x^{2}$ .

$\frac{1}{2} (x (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 3.2

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

$\frac{1}{2} (x (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 3.3

Replace all occurrences of $u_{1}$ with $64 - x^{2}$ .

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 4

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

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 5

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

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 7.2

Subtract $2$ from $1$ .

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$\frac{1}{2} (x (\frac{1}{2} {(64 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 8.2

Combine $\frac{1}{2}$ and ${(64 - x^{2})}^{- \frac{1}{2}}$ .

$\frac{1}{2} (x (\frac{{(64 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 8.3

Move ${(64 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} (x (\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 8.4

Combine $\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [64 - x^{2}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 9

By the Sum Rule, the derivative of $64 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [64] + \frac{d}{d x} [- x^{2}]$ .

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [64] + \frac{d}{d x} [- x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 10

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 11

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

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 12

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

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 13

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

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- (2 x)) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 14

Combine fractions.

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Step 14.1

Multiply $2$ by $- 1$ .

$\frac{1}{2} (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- 2 x) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 14.2

Combine $- 2$ and $\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{1}{2} (\frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}} x + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 14.3

Combine $\frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}}$ and $x$ .

$\frac{1}{2} (\frac{- 2 x \cdot x}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 15

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

$\frac{1}{2} (\frac{- 2 (x^{1} x)}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 16

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

$\frac{1}{2} (\frac{- 2 (x^{1} x^{1})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 17

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

$\frac{1}{2} (\frac{- 2 x^{1 + 1}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 18

Add $1$ and $1$ .

$\frac{1}{2} (\frac{- 2 x^{2}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 19

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

$\frac{1}{2} (\frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 20

Cancel the common factors.

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Step 20.1

Factor $2$ out of $2 {(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{1}{2} (\frac{2 (- x^{2})}{2 ({(64 - x^{2})}^{\frac{1}{2}})} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 20.2

Cancel the common factor.

$\frac{1}{2} (\frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 20.3

Rewrite the expression.

$\frac{1}{2} (\frac{- x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 21

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 22

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

$\frac{1}{2} (- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \cdot 1 + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 23

Multiply ${(64 - x^{2})}^{\frac{1}{2}}$ by $1$ .

$\frac{1}{2} (- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 24

To write ${(64 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{1}{2} (- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{{(64 - x^{2})}^{\frac{1}{2}} {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 25

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{- x^{2} + {(64 - x^{2})}^{\frac{1}{2}} {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 26

Multiply ${(64 - x^{2})}^{\frac{1}{2}}$ by ${(64 - x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 26.1

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

$\frac{1}{2} (\frac{- x^{2} + {(64 - x^{2})}^{\frac{1}{2} + \frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 26.2

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{- x^{2} + {(64 - x^{2})}^{\frac{1 + 1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 26.3

Add $1$ and $1$ .

$\frac{1}{2} (\frac{- x^{2} + {(64 - x^{2})}^{\frac{2}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 26.4

Divide $2$ by $2$ .

$\frac{1}{2} (\frac{- x^{2} + {(64 - x^{2})}^{1}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 27

Differentiate using the Constant Multiple Rule.

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Step 27.1

Simplify ${(64 - x^{2})}^{1}$ .

$\frac{1}{2} (\frac{- x^{2} + 64 - x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 27.2

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

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{d}{d x} [64 \arcsin (\frac{x}{8})])$

Step 27.3

Since $64$ is constant with respect to $x$ , the derivative of $64 \arcsin (\frac{x}{8})$ with respect to $x$ is $64 \frac{d}{d x} [\arcsin (\frac{x}{8})]$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + 64 \frac{d}{d x} [\arcsin (\frac{x}{8})])$

Step 28

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arcsin (x)$ and $g (x) = \frac{x}{8}$ .

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Step 28.1

To apply the Chain Rule, set $u_{2}$ as $\frac{x}{8}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + 64 (\frac{d}{d u_{2}} [\arcsin (u_{2})] \frac{d}{d x} [\frac{x}{8}]))$

Step 28.2

The derivative of $\arcsin (u_{2})$ with respect to $u_{2}$ is $\frac{1}{\sqrt{1 - {u_{2}}^{2}}}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + 64 (\frac{1}{\sqrt{1 - {u_{2}}^{2}}} \frac{d}{d x} [\frac{x}{8}]))$

Step 28.3

Replace all occurrences of $u_{2}$ with $\frac{x}{8}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + 64 (\frac{1}{\sqrt{1 - {(\frac{x}{8})}^{2}}} \frac{d}{d x} [\frac{x}{8}]))$

Step 29

Differentiate.

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Step 29.1

Combine $\frac{1}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ and $64$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{64}{\sqrt{1 - {(\frac{x}{8})}^{2}}} \frac{d}{d x} [\frac{x}{8}])$

Step 29.2

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{x}{8}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x]$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{64}{\sqrt{1 - {(\frac{x}{8})}^{2}}} (\frac{1}{8} \frac{d}{d x} [x]))$

Step 29.3

Simplify terms.

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Step 29.3.1

Multiply $\frac{1}{8}$ by $\frac{64}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{64}{8 \sqrt{1 - {(\frac{x}{8})}^{2}}} \frac{d}{d x} [x])$

Step 29.3.2

Cancel the common factor of $64$ and $8$ .

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Step 29.3.2.1

Factor $8$ out of $64$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{8 \cdot 8}{8 \sqrt{1 - {(\frac{x}{8})}^{2}}} \frac{d}{d x} [x])$

Step 29.3.2.2

Cancel the common factors.

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Step 29.3.2.2.1

Factor $8$ out of $8 \sqrt{1 - {(\frac{x}{8})}^{2}}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{8 \cdot 8}{8 (\sqrt{1 - {(\frac{x}{8})}^{2}})} \frac{d}{d x} [x])$

Step 29.3.2.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{8 \cdot 8}{8 \sqrt{1 - {(\frac{x}{8})}^{2}}} \frac{d}{d x} [x])$

Step 29.3.2.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}} \frac{d}{d x} [x])$

Step 29.4

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

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}} \cdot 1)$

Step 29.5

Multiply $\frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ by $1$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} + \frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}})$

Step 30

To write $\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{1 - {(\frac{x}{8})}^{2}}}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - {(\frac{x}{8})}^{2}}}{\sqrt{1 - {(\frac{x}{8})}^{2}}} + \frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}})$

Step 31

To write $\frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ as a fraction with a common denominator, multiply by $\frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{1}{2} (\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}} \cdot \frac{\sqrt{1 - {(\frac{x}{8})}^{2}}}{\sqrt{1 - {(\frac{x}{8})}^{2}}} + \frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}} \cdot \frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}})$

Step 32

Write each expression with a common denominator of $\sqrt{1 - {(\frac{x}{8})}^{2}} {(64 - x^{2})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 32.1

Multiply $\frac{- 2 x^{2} + 64}{{(64 - x^{2})}^{\frac{1}{2}}}$ by $\frac{\sqrt{1 - {(\frac{x}{8})}^{2}}}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ .

$\frac{1}{2} (\frac{(- 2 x^{2} + 64) \sqrt{1 - {(\frac{x}{8})}^{2}}}{{(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}} + \frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}} \cdot \frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}})$

Step 32.2

Multiply $\frac{8}{\sqrt{1 - {(\frac{x}{8})}^{2}}}$ by $\frac{{(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}}}$ .

$\frac{1}{2} (\frac{(- 2 x^{2} + 64) \sqrt{1 - {(\frac{x}{8})}^{2}}}{{(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}} + \frac{8 {(64 - x^{2})}^{\frac{1}{2}}}{\sqrt{1 - {(\frac{x}{8})}^{2}} {(64 - x^{2})}^{\frac{1}{2}}})$

Step 32.3

Reorder the factors of $\sqrt{1 - {(\frac{x}{8})}^{2}} {(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{1}{2} (\frac{(- 2 x^{2} + 64) \sqrt{1 - {(\frac{x}{8})}^{2}}}{{(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}} + \frac{8 {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}})$

Step 33

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{(- 2 x^{2} + 64) \sqrt{1 - {(\frac{x}{8})}^{2}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}}$

Step 34

Multiply $\frac{1}{2}$ by $\frac{(- 2 x^{2} + 64) \sqrt{1 - {(\frac{x}{8})}^{2}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{{(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{1 - {(\frac{x}{8})}^{2}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 ({(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}})}$

Step 35

Simplify.

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Step 35.1

Apply the product rule to $\frac{x}{8}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{1 - \frac{x^{2}}{8^{2}}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - {(\frac{x}{8})}^{2}}}$

Step 35.2

Apply the product rule to $\frac{x}{8}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{1 - \frac{x^{2}}{8^{2}}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3

Simplify the numerator.

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Step 35.3.1

Simplify each term.

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Step 35.3.1.1

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

$\frac{(- 2 x^{2} + 64) \sqrt{1^{2} - \frac{x^{2}}{8^{2}}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.2

Rewrite $\frac{x^{2}}{8^{2}}$ as ${(\frac{x}{8})}^{2}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{1^{2} - {(\frac{x}{8})}^{2}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{8}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{(1 + \frac{x}{8}) (1 - \frac{x}{8})} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.4

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

$\frac{(- 2 x^{2} + 64) \sqrt{(\frac{8}{8} + \frac{x}{8}) (1 - \frac{x}{8})} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.5

Combine the numerators over the common denominator.

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{8 + x}{8} (1 - \frac{x}{8})} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.6

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

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{8 + x}{8} (\frac{8}{8} - \frac{x}{8})} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.7

Combine the numerators over the common denominator.

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{8 + x}{8} \cdot \frac{8 - x}{8}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.8

Multiply $\frac{8 + x}{8}$ by $\frac{8 - x}{8}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{(8 + x) (8 - x)}{8 \cdot 8}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.9

Multiply $8$ by $8$ .

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{(8 + x) (8 - x)}{64}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.10

Rewrite $\frac{(8 + x) (8 - x)}{64}$ as ${(\frac{1}{8})}^{2} ((8 + x) (8 - x))$ .

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Step 35.3.1.10.1

Factor the perfect power $1^{2}$ out of $(8 + x) (8 - x)$ .

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{1^{2} ((8 + x) (8 - x))}{64}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.10.2

Factor the perfect power $8^{2}$ out of $64$ .

$\frac{(- 2 x^{2} + 64) \sqrt{\frac{1^{2} ((8 + x) (8 - x))}{8^{2} \cdot 1}} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.10.3

Rearrange the fraction $\frac{1^{2} ((8 + x) (8 - x))}{8^{2} \cdot 1}$ .

$\frac{(- 2 x^{2} + 64) \sqrt{{(\frac{1}{8})}^{2} ((8 + x) (8 - x))} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.11

Pull terms out from under the radical.

$\frac{(- 2 x^{2} + 64) (\frac{1}{8} \sqrt{(8 + x) (8 - x)}) + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.12

Combine $\frac{1}{8}$ and $\sqrt{(8 + x) (8 - x)}$ .

$\frac{(- 2 x^{2} + 64) \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.13

Apply the distributive property.

$\frac{- 2 x^{2} \frac{\sqrt{(8 + x) (8 - x)}}{8} + 64 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.14

Cancel the common factor of $2$ .

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Step 35.3.1.14.1

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

$\frac{2 (- x^{2}) \frac{\sqrt{(8 + x) (8 - x)}}{8} + 64 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.14.2

Factor $2$ out of $8$ .

$\frac{2 (- x^{2}) \frac{\sqrt{(8 + x) (8 - x)}}{2 (4)} + 64 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.14.3

Cancel the common factor.

$\frac{2 (- x^{2}) \frac{\sqrt{(8 + x) (8 - x)}}{2 \cdot 4} + 64 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.14.4

Rewrite the expression.

$\frac{- x^{2} \frac{\sqrt{(8 + x) (8 - x)}}{4} + 64 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.15

Combine $\frac{\sqrt{(8 + x) (8 - x)}}{4}$ and $x^{2}$ .

$\frac{- \frac{\sqrt{(8 + x) (8 - x)} x^{2}}{4} + 64 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.16

Cancel the common factor of $8$ .

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Step 35.3.1.16.1

Factor $8$ out of $64$ .

$\frac{- \frac{\sqrt{(8 + x) (8 - x)} x^{2}}{4} + 8 (8) \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.16.2

Cancel the common factor.

$\frac{- \frac{\sqrt{(8 + x) (8 - x)} x^{2}}{4} + 8 \cdot 8 \frac{\sqrt{(8 + x) (8 - x)}}{8} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.16.3

Rewrite the expression.

$\frac{- \frac{\sqrt{(8 + x) (8 - x)} x^{2}}{4} + 8 \sqrt{(8 + x) (8 - x)} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.17

To write $8 \sqrt{(8 + x) (8 - x)}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{- \frac{\sqrt{(8 + x) (8 - x)} x^{2}}{4} + 8 \sqrt{(8 + x) (8 - x)} \cdot \frac{4}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.18

Combine $8 \sqrt{(8 + x) (8 - x)}$ and $\frac{4}{4}$ .

$\frac{- \frac{\sqrt{(8 + x) (8 - x)} x^{2}}{4} + \frac{8 \sqrt{(8 + x) (8 - x)} \cdot 4}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.19

Combine the numerators over the common denominator.

$\frac{\frac{- \sqrt{(8 + x) (8 - x)} x^{2} + 8 \sqrt{(8 + x) (8 - x)} \cdot 4}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.20

Simplify the numerator.

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Step 35.3.1.20.1

Factor $\sqrt{(8 + x) (8 - x)}$ out of $- \sqrt{(8 + x) (8 - x)} x^{2} + 8 \sqrt{(8 + x) (8 - x)} \cdot 4$ .

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Step 35.3.1.20.1.1

Factor $\sqrt{(8 + x) (8 - x)}$ out of $- \sqrt{(8 + x) (8 - x)} x^{2}$ .

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2}) + 8 \sqrt{(8 + x) (8 - x)} \cdot 4}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.20.1.2

Factor $\sqrt{(8 + x) (8 - x)}$ out of $8 \sqrt{(8 + x) (8 - x)} \cdot 4$ .

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2}) + \sqrt{(8 + x) (8 - x)} (8 \cdot 4)}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.20.1.3

Factor $\sqrt{(8 + x) (8 - x)}$ out of $\sqrt{(8 + x) (8 - x)} (- x^{2}) + \sqrt{(8 + x) (8 - x)} (8 \cdot 4)$ .

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2} + 8 \cdot 4)}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.1.20.2

Multiply $8$ by $4$ .

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2} + 32)}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.2

To write $8 {(64 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2} + 32)}{4} + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot \frac{4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.3

Combine $8 {(64 - x^{2})}^{\frac{1}{2}}$ and $\frac{4}{4}$ .

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2} + 32)}{4} + \frac{8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.4

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{(8 + x) (8 - x)} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5

Simplify the numerator.

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Step 35.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(8 + x) (8 - x)}$ as ${((8 + x) (8 - x))}^{\frac{1}{2}}$ .

$\frac{\frac{{((8 + x) (8 - x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.2

Expand $(8 + x) (8 - x)$ using the FOIL Method.

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Step 35.3.5.2.1

Apply the distributive property.

$\frac{\frac{{(8 (8 - x) + x (8 - x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.2.2

Apply the distributive property.

$\frac{\frac{{(8 \cdot 8 + 8 (- x) + x (8 - x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.2.3

Apply the distributive property.

$\frac{\frac{{(8 \cdot 8 + 8 (- x) + x \cdot 8 + x (- x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3

Simplify and combine like terms.

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Step 35.3.5.3.1

Simplify each term.

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Step 35.3.5.3.1.1

Multiply $8$ by $8$ .

$\frac{\frac{{(64 + 8 (- x) + x \cdot 8 + x (- x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.1.2

Multiply $- 1$ by $8$ .

$\frac{\frac{{(64 - 8 x + x \cdot 8 + x (- x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.1.3

Move $8$ to the left of $x$ .

$\frac{\frac{{(64 - 8 x + 8 \cdot x + x (- x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{\frac{{(64 - 8 x + 8 x - x \cdot x)}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.1.5

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

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Step 35.3.5.3.1.5.1

Move $x$ .

$\frac{\frac{{(64 - 8 x + 8 x - (x \cdot x))}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.1.5.2

Multiply $x$ by $x$ .

$\frac{\frac{{(64 - 8 x + 8 x - x^{2})}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.2

Add $- 8 x$ and $8 x$ .

$\frac{\frac{{(64 + 0 - x^{2})}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.3.3

Add $64$ and $0$ .

$\frac{\frac{{(64 - x^{2})}^{\frac{1}{2}} (- x^{2} + 32) + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.4

Apply the distributive property.

$\frac{\frac{{(64 - x^{2})}^{\frac{1}{2}} (- x^{2}) + {(64 - x^{2})}^{\frac{1}{2}} \cdot 32 + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.5

Rewrite using the commutative property of multiplication.

$\frac{\frac{- {(64 - x^{2})}^{\frac{1}{2}} x^{2} + {(64 - x^{2})}^{\frac{1}{2}} \cdot 32 + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.6

Move $32$ to the left of ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{- {(64 - x^{2})}^{\frac{1}{2}} x^{2} + 32 \cdot {(64 - x^{2})}^{\frac{1}{2}} + 8 {(64 - x^{2})}^{\frac{1}{2}} \cdot 4}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.7

Multiply $4$ by $8$ .

$\frac{\frac{- {(64 - x^{2})}^{\frac{1}{2}} x^{2} + 32 {(64 - x^{2})}^{\frac{1}{2}} + 32 {(64 - x^{2})}^{\frac{1}{2}}}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.8

Add $32 {(64 - x^{2})}^{\frac{1}{2}}$ and $32 {(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{- {(64 - x^{2})}^{\frac{1}{2}} x^{2} + 64 {(64 - x^{2})}^{\frac{1}{2}}}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9

Rewrite $- {(64 - x^{2})}^{\frac{1}{2}} x^{2} + 64 {(64 - x^{2})}^{\frac{1}{2}}$ in a factored form.

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Step 35.3.5.9.1

Add parentheses.

$\frac{\frac{- ({(64 - x^{2})}^{\frac{1}{2}} x^{2}) + 64 {(64 - x^{2})}^{\frac{1}{2}}}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.2

Factor $u_{3}$ out of $- u_{3} x^{2} + 64 u_{3}$ .

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Step 35.3.5.9.2.1

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

$\frac{\frac{u_{3} (- x^{2}) + 64 u_{3}}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.2.2

Factor $u_{3}$ out of $64 u_{3}$ .

$\frac{\frac{u_{3} (- x^{2}) + u_{3} \cdot 64}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.2.3

Factor $u_{3}$ out of $u_{3} (- x^{2}) + u_{3} \cdot 64$ .

$\frac{\frac{u_{3} (- x^{2} + 64)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.3

Rewrite $64$ as $8^{2}$ .

$\frac{\frac{u_{3} (- x^{2} + 8^{2})}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.4

Reorder $- x^{2}$ and $8^{2}$ .

$\frac{\frac{u_{3} (8^{2} - x^{2})}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$\frac{\frac{u_{3} (8 + x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.6

Replace all occurrences of $u_{3}$ with ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.7

Simplify.

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Step 35.3.5.9.7.1

Apply the distributive property.

$\frac{\frac{({(64 - x^{2})}^{\frac{1}{2}} \cdot 8 + {(64 - x^{2})}^{\frac{1}{2}} x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.7.2

Move $8$ to the left of ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{(8 {(64 - x^{2})}^{\frac{1}{2}} + {(64 - x^{2})}^{\frac{1}{2}} x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.8

Let $u_{4} = {(64 - x^{2})}^{\frac{1}{2}}$ . Substitute $u_{4}$ for all occurrences of ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{(8 u_{4} + u_{4} x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.9

Factor $u_{4}$ out of $8 u_{4} + u_{4} x$ .

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Step 35.3.5.9.9.1

Factor $u_{4}$ out of $8 u_{4}$ .

$\frac{\frac{(u_{4} \cdot 8 + u_{4} x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.9.2

Factor $u_{4}$ out of $u_{4} x$ .

$\frac{\frac{(u_{4} \cdot 8 + u_{4} (x)) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.9.3

Factor $u_{4}$ out of $u_{4} \cdot 8 + u_{4} (x)$ .

$\frac{\frac{u_{4} (8 + x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.3.5.9.10

Replace all occurrences of $u_{4}$ with ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{8^{2}}}}$

Step 35.4

Combine terms.

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Step 35.4.1

Raise $8$ to the power of $2$ .

$\frac{\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}}}$

Step 35.4.2

Rewrite $\frac{\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4}}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}}}$ as a product.

$\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4} \cdot \frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}}}$

Step 35.4.3

Multiply $\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4}$ by $\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}}}$ .

$\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{4 (2 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}})}$

Step 35.4.4

Multiply $2$ by $4$ .

$\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{8 ({(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}})}$

Step 35.4.5

Cancel the common factor.

$\frac{{(64 - x^{2})}^{\frac{1}{2}} (8 + x) (8 - x)}{8 {(64 - x^{2})}^{\frac{1}{2}} \sqrt{1 - \frac{x^{2}}{64}}}$

Step 35.4.6

Rewrite the expression.

$\frac{(8 + x) (8 - x)}{8 \sqrt{1 - \frac{x^{2}}{64}}}$

Step 35.5

Simplify the denominator.

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Step 35.5.1

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

$\frac{(8 + x) (8 - x)}{8 \sqrt{\frac{64}{64} - \frac{x^{2}}{64}}}$

Step 35.5.2

Combine the numerators over the common denominator.

$\frac{(8 + x) (8 - x)}{8 \sqrt{\frac{64 - x^{2}}{64}}}$

Step 35.5.3

Rewrite $\frac{64 - x^{2}}{64}$ in a factored form.

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Step 35.5.3.1

Rewrite $64$ as $8^{2}$ .

$\frac{(8 + x) (8 - x)}{8 \sqrt{\frac{8^{2} - x^{2}}{64}}}$

Step 35.5.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$\frac{(8 + x) (8 - x)}{8 \sqrt{\frac{(8 + x) (8 - x)}{64}}}$

Step 35.5.4

Rewrite $\frac{(8 + x) (8 - x)}{64}$ as ${(\frac{1}{8})}^{2} ((8 + x) (8 - x))$ .

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Step 35.5.4.1

Factor the perfect power $1^{2}$ out of $(8 + x) (8 - x)$ .

$\frac{(8 + x) (8 - x)}{8 \sqrt{\frac{1^{2} ((8 + x) (8 - x))}{64}}}$

Step 35.5.4.2

Factor the perfect power $8^{2}$ out of $64$ .

$\frac{(8 + x) (8 - x)}{8 \sqrt{\frac{1^{2} ((8 + x) (8 - x))}{8^{2} \cdot 1}}}$

Step 35.5.4.3

Rearrange the fraction $\frac{1^{2} ((8 + x) (8 - x))}{8^{2} \cdot 1}$ .

$\frac{(8 + x) (8 - x)}{8 \sqrt{{(\frac{1}{8})}^{2} ((8 + x) (8 - x))}}$

Step 35.5.5

Pull terms out from under the radical.

$\frac{(8 + x) (8 - x)}{8 (\frac{1}{8} \sqrt{(8 + x) (8 - x)})}$

Step 35.5.6

Combine $\frac{1}{8}$ and $\sqrt{(8 + x) (8 - x)}$ .

$\frac{(8 + x) (8 - x)}{8 \frac{\sqrt{(8 + x) (8 - x)}}{8}}$

Step 35.6

Combine $8$ and $\frac{\sqrt{(8 + x) (8 - x)}}{8}$ .

$\frac{(8 + x) (8 - x)}{\frac{8 \sqrt{(8 + x) (8 - x)}}{8}}$

Step 35.7

Reduce the expression by cancelling the common factors.

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Step 35.7.1

Reduce the expression $\frac{8 \sqrt{(8 + x) (8 - x)}}{8}$ by cancelling the common factors.

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Step 35.7.1.1

Cancel the common factor.

$\frac{(8 + x) (8 - x)}{\frac{8 \sqrt{(8 + x) (8 - x)}}{8}}$

Step 35.7.1.2

Rewrite the expression.

$\frac{(8 + x) (8 - x)}{\frac{\sqrt{(8 + x) (8 - x)}}{1}}$

Step 35.7.2

Divide $\sqrt{(8 + x) (8 - x)}$ by $1$ .

$\frac{(8 + x) (8 - x)}{\sqrt{(8 + x) (8 - x)}}$

Step 35.8

Multiply $\frac{(8 + x) (8 - x)}{\sqrt{(8 + x) (8 - x)}}$ by $\frac{\sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)}}$ .

$\frac{(8 + x) (8 - x)}{\sqrt{(8 + x) (8 - x)}} \cdot \frac{\sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)}}$

Step 35.9

Combine and simplify the denominator.

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Step 35.9.1

Multiply $\frac{(8 + x) (8 - x)}{\sqrt{(8 + x) (8 - x)}}$ by $\frac{\sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)}}$ .

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)} \sqrt{(8 + x) (8 - x)}}$

Step 35.9.2

Raise $\sqrt{(8 + x) (8 - x)}$ to the power of $1$ .

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{1} \sqrt{(8 + x) (8 - x)}}$

Step 35.9.3

Raise $\sqrt{(8 + x) (8 - x)}$ to the power of $1$ .

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{1} {\sqrt{(8 + x) (8 - x)}}^{1}}$

Step 35.9.4

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

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{1 + 1}}$

Step 35.9.5

Add $1$ and $1$ .

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{2}}$

Step 35.9.6

Rewrite ${\sqrt{(8 + x) (8 - x)}}^{2}$ as $(8 + x) (8 - x)$ .

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Step 35.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(8 + x) (8 - x)}$ as ${((8 + x) (8 - x))}^{\frac{1}{2}}$ .

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{({((8 + x) (8 - x))}^{\frac{1}{2}})}^{2}}$

Step 35.9.6.2

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

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{\frac{1}{2} \cdot 2}}$

Step 35.9.6.3

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

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{\frac{2}{2}}}$

Step 35.9.6.4

Cancel the common factor of $2$ .

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Step 35.9.6.4.1

Cancel the common factor.

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{\frac{2}{2}}}$

Step 35.9.6.4.2

Rewrite the expression.

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{1}}$

Step 35.9.6.5

Simplify.

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{(8 + x) (8 - x)}$

Step 35.10

Cancel the common factor of $8 + x$ .

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Step 35.10.1

Cancel the common factor.

$\frac{(8 + x) (8 - x) \sqrt{(8 + x) (8 - x)}}{(8 + x) (8 - x)}$

Step 35.10.2

Rewrite the expression.

$\frac{(8 - x) \sqrt{(8 + x) (8 - x)}}{8 - x}$

Step 35.11

Cancel the common factor of $8 - x$ .

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Step 35.11.1

Cancel the common factor.

$\frac{(8 - x) \sqrt{(8 + x) (8 - x)}}{8 - x}$

Step 35.11.2

Divide $\sqrt{(8 + x) (8 - x)}$ by $1$ .

$\sqrt{(8 + x) (8 - x)}$