Calculus Examples

$\frac{1}{2} x^{2} \sqrt{16 - x^{2}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [\frac{x^{2}}{2} \cdot \sqrt{16 - x^{2}}]$

Step 1.2

Combine fractions.

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Step 1.2.1

Combine $\frac{x^{2}}{2}$ and $\sqrt{16 - x^{2}}$ .

$\frac{d}{d x} [\frac{x^{2} \sqrt{16 - x^{2}}}{2}]$

Step 1.2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 - x^{2}}$ as ${(16 - x^{2})}^{\frac{1}{2}}$ .

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

Step 1.3

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

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

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

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

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 $g (x) = 16 - x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $16 - x^{2}$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $16 - x^{2}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

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

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

Step 8.3

Move ${(16 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8.4

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

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

Step 9

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

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

Step 10

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Combine fractions.

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Step 14.1

Multiply $2$ by $- 1$ .

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

Step 14.2

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

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

Step 14.3

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

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

Step 15

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

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

Step 16

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

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

Step 17

Add $1$ and $2$ .

$\frac{1}{2} (\frac{- 2 x^{3}}{2 {(16 - x^{2})}^{\frac{1}{2}}} + {(16 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 18

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

$\frac{1}{2} (\frac{2 (- x^{3})}{2 {(16 - x^{2})}^{\frac{1}{2}}} + {(16 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 19

Cancel the common factors.

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Step 19.1

Factor $2$ out of $2 {(16 - x^{2})}^{\frac{1}{2}}$ .

$\frac{1}{2} (\frac{2 (- x^{3})}{2 ({(16 - x^{2})}^{\frac{1}{2}})} + {(16 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 19.2

Cancel the common factor.

$\frac{1}{2} (\frac{2 (- x^{3})}{2 {(16 - x^{2})}^{\frac{1}{2}}} + {(16 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 19.3

Rewrite the expression.

$\frac{1}{2} (\frac{- x^{3}}{{(16 - x^{2})}^{\frac{1}{2}}} + {(16 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 20

Move the negative in front of the fraction.

$\frac{1}{2} (- \frac{x^{3}}{{(16 - x^{2})}^{\frac{1}{2}}} + {(16 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x^{2}])$

Step 21

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

$\frac{1}{2} (- \frac{x^{3}}{{(16 - x^{2})}^{\frac{1}{2}}} + {(16 - x^{2})}^{\frac{1}{2}} (2 x))$

Step 22

Reorder.

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Step 22.1

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

$\frac{1}{2} (- \frac{x^{3}}{{(16 - x^{2})}^{\frac{1}{2}}} + 2 \cdot {(16 - x^{2})}^{\frac{1}{2}} x)$

Step 22.2

Move ${(- x^{2} + 16)}^{\frac{1}{2}}$ .

$\frac{1}{2} (- \frac{x^{3}}{{(- x^{2} + 16)}^{\frac{1}{2}}} + 2 x {(- x^{2} + 16)}^{\frac{1}{2}})$

Step 23

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

$\frac{1}{2} (- \frac{x^{3}}{{(- x^{2} + 16)}^{\frac{1}{2}}} + \frac{2 x {(- x^{2} + 16)}^{\frac{1}{2}} {(- x^{2} + 16)}^{\frac{1}{2}}}{{(- x^{2} + 16)}^{\frac{1}{2}}})$

Step 24

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x {(- x^{2} + 16)}^{\frac{1}{2}} {(- x^{2} + 16)}^{\frac{1}{2}}}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 25

Multiply ${(- x^{2} + 16)}^{\frac{1}{2}}$ by ${(- x^{2} + 16)}^{\frac{1}{2}}$ by adding the exponents.

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Step 25.1

Move ${(- x^{2} + 16)}^{\frac{1}{2}}$ .

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x ({(- x^{2} + 16)}^{\frac{1}{2}} {(- x^{2} + 16)}^{\frac{1}{2}})}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 25.2

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

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x {(- x^{2} + 16)}^{\frac{1}{2} + \frac{1}{2}}}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 25.3

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x {(- x^{2} + 16)}^{\frac{1 + 1}{2}}}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 25.4

Add $1$ and $1$ .

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x {(- x^{2} + 16)}^{\frac{2}{2}}}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 25.5

Divide $2$ by $2$ .

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x {(- x^{2} + 16)}^{1}}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 26

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

$\frac{1}{2} \cdot \frac{- x^{3} + 2 x (- x^{2} + 16)}{{(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 27

Multiply $\frac{1}{2}$ by $\frac{- x^{3} + 2 x (- x^{2} + 16)}{{(- x^{2} + 16)}^{\frac{1}{2}}}$ .

$\frac{- x^{3} + 2 x (- x^{2} + 16)}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28

Simplify.

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Step 28.1

Apply the distributive property.

$\frac{- x^{3} + 2 x (- x^{2}) + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2

Simplify the numerator.

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Step 28.2.1

Simplify each term.

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Step 28.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{- x^{3} + 2 \cdot - 1 x \cdot x^{2} + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.1.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 28.2.1.2.1

Move $x^{2}$ .

$\frac{- x^{3} + 2 \cdot - 1 (x^{2} x) + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.1.2.2

Multiply $x^{2}$ by $x$ .

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Step 28.2.1.2.2.1

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

$\frac{- x^{3} + 2 \cdot - 1 (x^{2} x^{1}) + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.1.2.2.2

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

$\frac{- x^{3} + 2 \cdot - 1 x^{2 + 1} + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.1.2.3

Add $2$ and $1$ .

$\frac{- x^{3} + 2 \cdot - 1 x^{3} + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.1.3

Multiply $2$ by $- 1$ .

$\frac{- x^{3} - 2 x^{3} + 2 x \cdot 16}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.1.4

Multiply $16$ by $2$ .

$\frac{- x^{3} - 2 x^{3} + 32 x}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.2.2

Subtract $2 x^{3}$ from $- x^{3}$ .

$\frac{- 3 x^{3} + 32 x}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.3

Factor $x$ out of $- 3 x^{3} + 32 x$ .

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Step 28.3.1

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

$\frac{x (- 3 x^{2}) + 32 x}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.3.2

Factor $x$ out of $32 x$ .

$\frac{x (- 3 x^{2}) + x \cdot 32}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.3.3

Factor $x$ out of $x (- 3 x^{2}) + x \cdot 32$ .

$\frac{x (- 3 x^{2} + 32)}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.4

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{x (- (3 x^{2}) + 32)}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.5

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

$\frac{x (- (3 x^{2}) - 1 (- 32))}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.6

Factor $- 1$ out of $- (3 x^{2}) - 1 (- 32)$ .

$\frac{x (- (3 x^{2} - 32))}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.7

Rewrite $- (3 x^{2} - 32)$ as $- 1 (3 x^{2} - 32)$ .

$\frac{x (- 1 (3 x^{2} - 32))}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$

Step 28.8

Move the negative in front of the fraction.

$- \frac{x (3 x^{2} - 32)}{2 {(- x^{2} + 16)}^{\frac{1}{2}}}$