Calculus Examples

f (x) = \frac{1}{x^{2}}

$f (x) = \frac{1}{x^{2}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite

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

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

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

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

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

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

Step 1.2

Multiply the exponents in

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

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

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Step 1.2.1

Apply the power rule and multiply exponents,

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

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

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

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

Step 1.2.2

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

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

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

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

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

Step 2

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

$n = - 2$ .

- 2 x^{- 3}

$- 2 x^{- 3}$

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule

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

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

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

$- 2 \frac{1}{x^{3}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine

- 2

$- 2$ and

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

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

\frac{- 2}{x^{3}}

$\frac{- 2}{x^{3}}$

Step 3.2.2

Move the negative in front of the fraction.

- \frac{2}{x^{3}}

$- \frac{2}{x^{3}}$

- \frac{2}{x^{3}}

$- \frac{2}{x^{3}}$

- \frac{2}{x^{3}}

$- \frac{2}{x^{3}}$