Calculus Examples

$y = - 9 {(6 x^{2} + 4)}^{- 6}$

Step 1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 {(6 x^{2} + 4)}^{- 6}$ with respect to $x$ is $- 9 \frac{d}{d x} [{(6 x^{2} + 4)}^{- 6}]$ .

$- 9 \frac{d}{d x} [{(6 x^{2} + 4)}^{- 6}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 6}$ and $g (x) = 6 x^{2} + 4$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x^{2} + 4$ .

$- 9 (\frac{d}{d u} [u^{- 6}] \frac{d}{d x} [6 x^{2} + 4])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 6$ .

$- 9 (- 6 u^{- 7} \frac{d}{d x} [6 x^{2} + 4])$

Step 2.3

Replace all occurrences of $u$ with $6 x^{2} + 4$ .

$- 9 (- 6 {(6 x^{2} + 4)}^{- 7} \frac{d}{d x} [6 x^{2} + 4])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 6$ by $- 9$ .

$54 ({(6 x^{2} + 4)}^{- 7} \frac{d}{d x} [6 x^{2} + 4])$

Step 3.2

By the Sum Rule, the derivative of $6 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [4]$ .

$54 {(6 x^{2} + 4)}^{- 7} (\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [4])$

Step 3.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$54 {(6 x^{2} + 4)}^{- 7} (6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])$

Step 3.4

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

$54 {(6 x^{2} + 4)}^{- 7} (6 (2 x) + \frac{d}{d x} [4])$

Step 3.5

Multiply $2$ by $6$ .

$54 {(6 x^{2} + 4)}^{- 7} (12 x + \frac{d}{d x} [4])$

Step 3.6

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

$54 {(6 x^{2} + 4)}^{- 7} (12 x + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $12 x$ and $0$ .

$54 {(6 x^{2} + 4)}^{- 7} (12 x)$

Step 3.7.2

Multiply $12$ by $54$ .

$648 {(6 x^{2} + 4)}^{- 7} x$

Step 4

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

$648 \frac{1}{{(6 x^{2} + 4)}^{7}} x$

Step 5

Combine terms.

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Step 5.1

Combine $648$ and $\frac{1}{{(6 x^{2} + 4)}^{7}}$ .

$\frac{648}{{(6 x^{2} + 4)}^{7}} x$

Step 5.2

Combine $\frac{648}{{(6 x^{2} + 4)}^{7}}$ and $x$ .

$\frac{648 x}{{(6 x^{2} + 4)}^{7}}$