Calculus Examples

$4 x \sqrt{64 - x^{2}}$

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 $\sqrt{64 - x^{2}}$ as ${(64 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 x {(64 - x^{2})}^{\frac{1}{2}}]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x {(64 - x^{2})}^{\frac{1}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}}]$ .

$4 \frac{d}{d x} [x {(64 - x^{2})}^{\frac{1}{2}}]$

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

$4 (x \frac{d}{d x} [{(64 - x^{2})}^{\frac{1}{2}}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

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 $u$ as $64 - x^{2}$ .

$4 (x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 3.2

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

$4 (x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 3.3

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

$4 (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 4

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

$4 (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 5

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

$4 (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 6

Combine the numerators over the common denominator.

$4 (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$4 (x (\frac{1}{2} {(64 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 7.2

Subtract $2$ from $1$ .

$4 (x (\frac{1}{2} {(64 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$4 (x (\frac{1}{2} {(64 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.2

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

$4 (x (\frac{{(64 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.3

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

$4 (x (\frac{1}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [64 - x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 8.4

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

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [64 - x^{2}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 9

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

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [64] + \frac{d}{d x} [- x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 10

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

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 11

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

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}] + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 12

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

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}]) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 13

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

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- (2 x)) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 14

Combine fractions.

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Step 14.1

Multiply $2$ by $- 1$ .

$4 (\frac{x}{2 {(64 - x^{2})}^{\frac{1}{2}}} (- 2 x) + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 14.2

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

$4 (\frac{- 2 x}{2 {(64 - x^{2})}^{\frac{1}{2}}} x + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 14.3

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

$4 (\frac{- 2 x \cdot x}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 15

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

$4 (\frac{- 2 (x^{1} x)}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 16

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

$4 (\frac{- 2 (x^{1} x^{1})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 17

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

$4 (\frac{- 2 x^{1 + 1}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 18

Add $1$ and $1$ .

$4 (\frac{- 2 x^{2}}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 19

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

$4 (\frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 20

Cancel the common factors.

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Step 20.1

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

$4 (\frac{2 (- x^{2})}{2 ({(64 - x^{2})}^{\frac{1}{2}})} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 20.2

Cancel the common factor.

$4 (\frac{2 (- x^{2})}{2 {(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 20.3

Rewrite the expression.

$4 (\frac{- x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 21

Move the negative in front of the fraction.

$4 (- \frac{x^{2}}{{(64 - x^{2})}^{\frac{1}{2}}} + {(64 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x])$

Step 22

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

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

Step 23

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

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

Step 24

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

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

Step 25

Combine the numerators over the common denominator.

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

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.

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

Step 26.2

Combine the numerators over the common denominator.

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

Step 26.3

Add $1$ and $1$ .

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

Step 26.4

Divide $2$ by $2$ .

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

Step 27

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

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

Step 28

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

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

Step 29

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

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

Step 30

Simplify.

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Step 30.1

Apply the distributive property.

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

Step 30.2

Simplify each term.

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Step 30.2.1

Multiply $- 2$ by $4$ .

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

Step 30.2.2

Multiply $4$ by $64$ .

$\frac{- 8 x^{2} + 256}{{(64 - x^{2})}^{\frac{1}{2}}}$

Step 30.3

Reorder terms.

$\frac{- 8 x^{2} + 256}{{(- x^{2} + 64)}^{\frac{1}{2}}}$

Step 30.4

Factor $8$ out of $- 8 x^{2} + 256$ .

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Step 30.4.1

Factor $8$ out of $- 8 x^{2}$ .

$\frac{8 (- x^{2}) + 256}{{(- x^{2} + 64)}^{\frac{1}{2}}}$

Step 30.4.2

Factor $8$ out of $256$ .

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

Step 30.4.3

Factor $8$ out of $8 (- x^{2}) + 8 (32)$ .

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

Step 30.5

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

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

Step 30.6

Rewrite $32$ as $- 1 (- 32)$ .

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

Step 30.7

Factor $- 1$ out of $- (x^{2}) - 1 (- 32)$ .

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

Step 30.8

Rewrite $- (x^{2} - 32)$ as $- 1 (x^{2} - 32)$ .

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

Step 30.9

Move the negative in front of the fraction.

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