Calculus Examples

$\frac{2}{- x^{2} - 9}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{- x^{2} - 9}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{- x^{2} - 9}]$ .

$2 \frac{d}{d x} [\frac{1}{- x^{2} - 9}]$

Step 1.2

Rewrite $\frac{1}{- x^{2} - 9}$ as ${(- x^{2} - 9)}^{- 1}$ .

$2 \frac{d}{d x} [{(- x^{2} - 9)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = - x^{2} - 9$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- x^{2} - 9$ .

$2 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [- x^{2} - 9])$

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- x^{2} - 9$ .

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

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

By the Sum Rule, the derivative of $- x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 9]$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $2$ by $- 1$ .

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

Step 3.6

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$- 2 {(- x^{2} - 9)}^{- 2} (- 2 x + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $- 2 x$ and $0$ .

$- 2 {(- x^{2} - 9)}^{- 2} (- 2 x)$

Step 3.7.2

Multiply $- 2$ by $- 2$ .

$4 {(- x^{2} - 9)}^{- 2} x$

Step 4

Simplify.

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Step 4.1

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

$4 \frac{1}{{(- x^{2} - 9)}^{2}} x$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $4$ and $\frac{1}{{(- x^{2} - 9)}^{2}}$ .

$\frac{4}{{(- x^{2} - 9)}^{2}} x$

Step 4.2.2

Combine $\frac{4}{{(- x^{2} - 9)}^{2}}$ and $x$ .

$\frac{4 x}{{(- x^{2} - 9)}^{2}}$