Calculus Examples

$f (x) = - 7 {(3 x^{4} + 2)}^{- 4}$

Step 1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 {(3 x^{4} + 2)}^{- 4}$ with respect to $x$ is $- 7 \frac{d}{d x} [{(3 x^{4} + 2)}^{- 4}]$ .

$- 7 \frac{d}{d x} [{(3 x^{4} + 2)}^{- 4}]$

Step 2

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

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Step 2.1

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

$- 7 (\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [3 x^{4} + 2])$

Step 2.2

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

$- 7 (- 4 u^{- 5} \frac{d}{d x} [3 x^{4} + 2])$

Step 2.3

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

$- 7 (- 4 {(3 x^{4} + 2)}^{- 5} \frac{d}{d x} [3 x^{4} + 2])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 4$ by $- 7$ .

$28 ({(3 x^{4} + 2)}^{- 5} \frac{d}{d x} [3 x^{4} + 2])$

Step 3.2

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

$28 {(3 x^{4} + 2)}^{- 5} (\frac{d}{d x} [3 x^{4}] + \frac{d}{d x} [2])$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{4}$ with respect to $x$ is $3 \frac{d}{d x} [x^{4}]$ .

$28 {(3 x^{4} + 2)}^{- 5} (3 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [2])$

Step 3.4

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

$28 {(3 x^{4} + 2)}^{- 5} (3 (4 x^{3}) + \frac{d}{d x} [2])$

Step 3.5

Multiply $4$ by $3$ .

$28 {(3 x^{4} + 2)}^{- 5} (12 x^{3} + \frac{d}{d x} [2])$

Step 3.6

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$28 {(3 x^{4} + 2)}^{- 5} (12 x^{3} + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $12 x^{3}$ and $0$ .

$28 {(3 x^{4} + 2)}^{- 5} (12 x^{3})$

Step 3.7.2

Multiply $12$ by $28$ .

$336 {(3 x^{4} + 2)}^{- 5} x^{3}$

Step 4

Simplify.

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Step 4.1

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

$336 \frac{1}{{(3 x^{4} + 2)}^{5}} x^{3}$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $336$ and $\frac{1}{{(3 x^{4} + 2)}^{5}}$ .

$\frac{336}{{(3 x^{4} + 2)}^{5}} x^{3}$

Step 4.2.2

Combine $\frac{336}{{(3 x^{4} + 2)}^{5}}$ and $x^{3}$ .

$\frac{336 x^{3}}{{(3 x^{4} + 2)}^{5}}$