Calculus Examples

\frac{4}{x^{2}}

$\frac{4}{x^{2}}$

Step 1

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

\frac{4}{x^{2}}

$\frac{4}{x^{2}}$ with respect to

x

$x$ is

4 \frac{d}{d x} [\frac{1}{x^{2}}]

$4 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

4 \frac{d}{d x} [\frac{1}{x^{2}}]

$4 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite

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

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

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

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

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

$4 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.2

Multiply the exponents in

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

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

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Step 2.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

4 \frac{d}{d x} [x^{2 \cdot - 1}]

$4 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2

Multiply

2

$2$ by

- 1

$- 1$ .

4 \frac{d}{d x} [x^{- 2}]

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

4 \frac{d}{d x} [x^{- 2}]

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

4 \frac{d}{d x} [x^{- 2}]

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

Step 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

$n = - 2$ .

4 (- 2 x^{- 3})

$4 (- 2 x^{- 3})$

Step 4

Multiply

- 2

$- 2$ by

4

$4$ .

- 8 x^{- 3}

$- 8 x^{- 3}$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule

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

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

- 8 \frac{1}{x^{3}}

$- 8 \frac{1}{x^{3}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine

- 8

$- 8$ and

\frac{1}{x^{3}}

$\frac{1}{x^{3}}$ .

\frac{- 8}{x^{3}}

$\frac{- 8}{x^{3}}$

Step 5.2.2

Move the negative in front of the fraction.

- \frac{8}{x^{3}}

$- \frac{8}{x^{3}}$

- \frac{8}{x^{3}}

$- \frac{8}{x^{3}}$

- \frac{8}{x^{3}}

$- \frac{8}{x^{3}}$