Calculus Examples

$y = {(4 x^{2} - 13)}^{- 14}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 14}$ and $g (x) = 4 x^{2} - 13$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 13$ .

$\frac{d}{d u} [u^{- 14}] \frac{d}{d x} [4 x^{2} - 13]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 14$ .

$- 14 u^{- 15} \frac{d}{d x} [4 x^{2} - 13]$

Step 1.3

Replace all occurrences of $u$ with $4 x^{2} - 13$ .

$- 14 {(4 x^{2} - 13)}^{- 15} \frac{d}{d x} [4 x^{2} - 13]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{2} - 13$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 13]$ .

$- 14 {(4 x^{2} - 13)}^{- 15} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 13])$

Step 2.2

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

$- 14 {(4 x^{2} - 13)}^{- 15} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 13])$

Step 2.3

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

$- 14 {(4 x^{2} - 13)}^{- 15} (4 (2 x) + \frac{d}{d x} [- 13])$

Step 2.4

Multiply $2$ by $4$ .

$- 14 {(4 x^{2} - 13)}^{- 15} (8 x + \frac{d}{d x} [- 13])$

Step 2.5

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13$ with respect to $x$ is $0$ .

$- 14 {(4 x^{2} - 13)}^{- 15} (8 x + 0)$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $8 x$ and $0$ .

$- 14 {(4 x^{2} - 13)}^{- 15} (8 x)$

Step 2.6.2

Multiply $8$ by $- 14$ .

$- 112 {(4 x^{2} - 13)}^{- 15} x$

Step 3

Simplify.

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Step 3.1

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

$- 112 \frac{1}{{(4 x^{2} - 13)}^{15}} x$

Step 3.2

Combine terms.

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Step 3.2.1

Combine $- 112$ and $\frac{1}{{(4 x^{2} - 13)}^{15}}$ .

$\frac{- 112}{{(4 x^{2} - 13)}^{15}} x$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{112}{{(4 x^{2} - 13)}^{15}} x$

Step 3.2.3

Combine $x$ and $\frac{112}{{(4 x^{2} - 13)}^{15}}$ .

$- \frac{x \cdot 112}{{(4 x^{2} - 13)}^{15}}$

Step 3.2.4

Move $112$ to the left of $x$ .

$- \frac{112 x}{{(4 x^{2} - 13)}^{15}}$