Calculus Examples

\frac{5}{x^{2} + 5}

$\frac{5}{x^{2} + 5}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

\frac{5}{x^{2} + 5}

$\frac{5}{x^{2} + 5}$ with respect to

x

$x$ is

5 \frac{d}{d x} [\frac{1}{x^{2} + 5}]

$5 \frac{d}{d x} [\frac{1}{x^{2} + 5}]$ .

5 \frac{d}{d x} [\frac{1}{x^{2} + 5}]

$5 \frac{d}{d x} [\frac{1}{x^{2} + 5}]$

Step 1.2

Rewrite

\frac{1}{x^{2} + 5}

$\frac{1}{x^{2} + 5}$ as

{(x^{2} + 5)}^{- 1}

${(x^{2} + 5)}^{- 1}$ .

5 \frac{d}{d x} [{(x^{2} + 5)}^{- 1}]

$5 \frac{d}{d x} [{(x^{2} + 5)}^{- 1}]$

5 \frac{d}{d x} [{(x^{2} + 5)}^{- 1}]

$5 \frac{d}{d x} [{(x^{2} + 5)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{- 1}

$f (x) = x^{- 1}$ and

g (x) = x^{2} + 5

$g (x) = x^{2} + 5$ .

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Step 2.1

To apply the Chain Rule, set

u

$u$ as

x^{2} + 5

$x^{2} + 5$ .

5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 5])

$5 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 5])$

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = - 1

$n = - 1$ .

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

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

Step 2.3

Replace all occurrences of

u

$u$ with

x^{2} + 5

$x^{2} + 5$ .

5 (- {(x^{2} + 5)}^{- 2} \frac{d}{d x} [x^{2} + 5])

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

5 (- {(x^{2} + 5)}^{- 2} \frac{d}{d x} [x^{2} + 5])

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

Step 3

Differentiate.

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Step 3.1

Multiply

- 1

$- 1$ by

5

$5$ .

- 5 ({(x^{2} + 5)}^{- 2} \frac{d}{d x} [x^{2} + 5])

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

Step 3.2

By the Sum Rule, the derivative of

x^{2} + 5

$x^{2} + 5$ with respect to

x

$x$ is

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5]$ .

- 5 {(x^{2} + 5)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [5])

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

Step 3.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

- 5 {(x^{2} + 5)}^{- 2} (2 x + \frac{d}{d x} [5])

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

Step 3.4

Since

5

$5$ is constant with respect to

x

$x$ , the derivative of

5

$5$ with respect to

x

$x$ is

0

$0$ .

- 5 {(x^{2} + 5)}^{- 2} (2 x + 0)

$- 5 {(x^{2} + 5)}^{- 2} (2 x + 0)$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add

2 x

$2 x$ and

0

$0$ .

- 5 {(x^{2} + 5)}^{- 2} (2 x)

$- 5 {(x^{2} + 5)}^{- 2} (2 x)$

Step 3.5.2

Multiply

2

$2$ by

- 5

$- 5$ .

- 10 {(x^{2} + 5)}^{- 2} x

$- 10 {(x^{2} + 5)}^{- 2} x$

- 10 {(x^{2} + 5)}^{- 2} x

$- 10 {(x^{2} + 5)}^{- 2} x$

- 10 {(x^{2} + 5)}^{- 2} x

$- 10 {(x^{2} + 5)}^{- 2} x$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

- 10 \frac{1}{{(x^{2} + 5)}^{2}} x

$- 10 \frac{1}{{(x^{2} + 5)}^{2}} x$

Step 4.2

Combine terms.

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Step 4.2.1

Combine

- 10

$- 10$ and

\frac{1}{{(x^{2} + 5)}^{2}}

$\frac{1}{{(x^{2} + 5)}^{2}}$ .

\frac{- 10}{{(x^{2} + 5)}^{2}} x

$\frac{- 10}{{(x^{2} + 5)}^{2}} x$

Step 4.2.2

Move the negative in front of the fraction.

- \frac{10}{{(x^{2} + 5)}^{2}} x

$- \frac{10}{{(x^{2} + 5)}^{2}} x$

Step 4.2.3

Combine

x

$x$ and

\frac{10}{{(x^{2} + 5)}^{2}}

$\frac{10}{{(x^{2} + 5)}^{2}}$ .

- \frac{x \cdot 10}{{(x^{2} + 5)}^{2}}

$- \frac{x \cdot 10}{{(x^{2} + 5)}^{2}}$

Step 4.2.4

Move

10

$10$ to the left of

x

$x$ .

- \frac{10 x}{{(x^{2} + 5)}^{2}}

$- \frac{10 x}{{(x^{2} + 5)}^{2}}$

- \frac{10 x}{{(x^{2} + 5)}^{2}}

$- \frac{10 x}{{(x^{2} + 5)}^{2}}$

- \frac{10 x}{{(x^{2} + 5)}^{2}}

$- \frac{10 x}{{(x^{2} + 5)}^{2}}$