Calculus Examples

$f (x) = {(9 - x^{2})}^{\frac{2}{3}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = 9 - x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $9 - x^{2}$ .

$\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [9 - x^{2}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [9 - x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $9 - x^{2}$ .

$\frac{2}{3} {(9 - x^{2})}^{\frac{2}{3} - 1} \frac{d}{d x} [9 - x^{2}]$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} {(9 - x^{2})}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [9 - x^{2}]$

Step 3

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{2}{3} {(9 - x^{2})}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [9 - x^{2}]$

Step 4

Combine the numerators over the common denominator.

$\frac{2}{3} {(9 - x^{2})}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [9 - x^{2}]$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} {(9 - x^{2})}^{\frac{2 - 3}{3}} \frac{d}{d x} [9 - x^{2}]$

Step 5.2

Subtract $3$ from $2$ .

$\frac{2}{3} {(9 - x^{2})}^{\frac{- 1}{3}} \frac{d}{d x} [9 - x^{2}]$

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

$\frac{2}{3} {(9 - x^{2})}^{- \frac{1}{3}} \frac{d}{d x} [9 - x^{2}]$

Step 6.2

Combine $\frac{2}{3}$ and ${(9 - x^{2})}^{- \frac{1}{3}}$ .

$\frac{2 {(9 - x^{2})}^{- \frac{1}{3}}}{3} \frac{d}{d x} [9 - x^{2}]$

Step 6.3

Move ${(9 - x^{2})}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} \frac{d}{d x} [9 - x^{2}]$

Step 7

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} (\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}])$

Step 8

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} (0 + \frac{d}{d x} [- x^{2}])$

Step 9

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

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} \frac{d}{d x} [- x^{2}]$

Step 10

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

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} (- \frac{d}{d x} [x^{2}])$

Step 11

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

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} (- (2 x))$

Step 12

Combine fractions.

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Step 12.1

Multiply $2$ by $- 1$ .

$\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}} (- 2 x)$

Step 12.2

Combine $- 2$ and $\frac{2}{3 {(9 - x^{2})}^{\frac{1}{3}}}$ .

$\frac{- 2 \cdot 2}{3 {(9 - x^{2})}^{\frac{1}{3}}} x$

Step 12.3

Multiply $- 2$ by $2$ .

$\frac{- 4}{3 {(9 - x^{2})}^{\frac{1}{3}}} x$

Step 12.4

Combine $\frac{- 4}{3 {(9 - x^{2})}^{\frac{1}{3}}}$ and $x$ .

$\frac{- 4 x}{3 {(9 - x^{2})}^{\frac{1}{3}}}$

Step 12.5

Move the negative in front of the fraction.

$- \frac{4 x}{3 {(9 - x^{2})}^{\frac{1}{3}}}$