Calculus Examples

${(- 3 + 2 x^{4})}^{- 2}$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3

Evaluate $\frac{d}{d x} [2 x^{4}]$ .

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- 2 {(- 3 + 2 x^{4})}^{- 3} (0 + 2 (4 x^{3}))$

Step 3.3

Multiply $4$ by $2$ .

$- 2 {(- 3 + 2 x^{4})}^{- 3} (0 + 8 x^{3})$

Step 4

Add $0$ and $8 x^{3}$ .

$- 2 {(- 3 + 2 x^{4})}^{- 3} (8 x^{3})$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Combine terms.

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Step 5.2.1

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

$\frac{- 2}{{(- 3 + 2 x^{4})}^{3}} (8 x^{3})$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{2}{{(- 3 + 2 x^{4})}^{3}} (8 x^{3})$

Step 5.2.3

Multiply $8$ by $- 1$ .

$- 8 \frac{2}{{(- 3 + 2 x^{4})}^{3}} x^{3}$

Step 5.2.4

Combine $- 8$ and $\frac{2}{{(- 3 + 2 x^{4})}^{3}}$ .

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

Step 5.2.5

Multiply $- 8$ by $2$ .

$\frac{- 16}{{(- 3 + 2 x^{4})}^{3}} x^{3}$

Step 5.2.6

Combine $\frac{- 16}{{(- 3 + 2 x^{4})}^{3}}$ and $x^{3}$ .

$\frac{- 16 x^{3}}{{(- 3 + 2 x^{4})}^{3}}$

Step 5.2.7

Move the negative in front of the fraction.

$- \frac{16 x^{3}}{{(- 3 + 2 x^{4})}^{3}}$