Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 4 - \frac{1}{3} x^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 - \frac{1}{3} x^{2}$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [4 - \frac{1}{3} x^{2}]$

Step 1.2

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

$3 u^{2} \frac{d}{d x} [4 - \frac{1}{3} x^{2}]$

Step 1.3

Replace all occurrences of $u$ with $4 - \frac{1}{3} x^{2}$ .

$3 {(4 - \frac{1}{3} x^{2})}^{2} \frac{d}{d x} [4 - \frac{1}{3} x^{2}]$

Step 2

Differentiate.

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Step 2.1

Combine $x^{2}$ and $\frac{1}{3}$ .

$3 {(4 - \frac{x^{2}}{3})}^{2} \frac{d}{d x} [4 - \frac{1}{3} x^{2}]$

Step 2.2

Combine $x^{2}$ and $\frac{1}{3}$ .

$3 {(4 - \frac{x^{2}}{3})}^{2} \frac{d}{d x} [4 - \frac{x^{2}}{3}]$

Step 2.3

By the Sum Rule, the derivative of $4 - \frac{x^{2}}{3}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- \frac{x^{2}}{3}]$ .

$3 {(4 - \frac{x^{2}}{3})}^{2} (\frac{d}{d x} [4] + \frac{d}{d x} [- \frac{x^{2}}{3}])$

Step 2.4

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$3 {(4 - \frac{x^{2}}{3})}^{2} (0 + \frac{d}{d x} [- \frac{x^{2}}{3}])$

Step 2.5

Add $0$ and $\frac{d}{d x} [- \frac{x^{2}}{3}]$ .

$3 {(4 - \frac{x^{2}}{3})}^{2} \frac{d}{d x} [- \frac{x^{2}}{3}]$

Step 2.6

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [x^{2}]$ .

$3 {(4 - \frac{x^{2}}{3})}^{2} (- \frac{1}{3} \frac{d}{d x} [x^{2}])$

Step 2.7

Simplify terms.

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Step 2.7.1

Multiply $- 1$ by $3$ .

$- 3 {(4 - \frac{x^{2}}{3})}^{2} (\frac{1}{3} \frac{d}{d x} [x^{2}])$

Step 2.7.2

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

$\frac{- 3}{3} {(4 - \frac{x^{2}}{3})}^{2} \frac{d}{d x} [x^{2}]$

Step 2.7.3

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

$\frac{- 3 {(4 - \frac{x^{2}}{3})}^{2}}{3} \frac{d}{d x} [x^{2}]$

Step 2.7.4

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

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Step 2.7.4.1

Factor $3$ out of $- 3 {(4 - \frac{x^{2}}{3})}^{2}$ .

$\frac{3 (- {(4 - \frac{x^{2}}{3})}^{2})}{3} \frac{d}{d x} [x^{2}]$

Step 2.7.4.2

Cancel the common factors.

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Step 2.7.4.2.1

Factor $3$ out of $3$ .

$\frac{3 (- {(4 - \frac{x^{2}}{3})}^{2})}{3 (1)} \frac{d}{d x} [x^{2}]$

Step 2.7.4.2.2

Cancel the common factor.

$\frac{3 (- {(4 - \frac{x^{2}}{3})}^{2})}{3 \cdot 1} \frac{d}{d x} [x^{2}]$

Step 2.7.4.2.3

Rewrite the expression.

$\frac{- {(4 - \frac{x^{2}}{3})}^{2}}{1} \frac{d}{d x} [x^{2}]$

Step 2.7.4.2.4

Divide $- {(4 - \frac{x^{2}}{3})}^{2}$ by $1$ .

$- {(4 - \frac{x^{2}}{3})}^{2} \frac{d}{d x} [x^{2}]$

Step 2.8

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

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

Step 2.9

Simplify the expression.

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Step 2.9.1

Multiply $2$ by $- 1$ .

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

Step 2.9.2

Reorder the factors of $- 2 {(4 - \frac{x^{2}}{3})}^{2} x$ .

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