Calculus Examples

\frac{1}{x^{6}}

$\frac{1}{x^{6}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite

\frac{1}{x^{6}}

$\frac{1}{x^{6}}$ as

{(x^{6})}^{- 1}

${(x^{6})}^{- 1}$ .

\frac{d}{d x} [{(x^{6})}^{- 1}]

$\frac{d}{d x} [{(x^{6})}^{- 1}]$

Step 1.2

Multiply the exponents in

{(x^{6})}^{- 1}

${(x^{6})}^{- 1}$ .

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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^{6 \cdot - 1}]

$\frac{d}{d x} [x^{6 \cdot - 1}]$

Step 1.2.2

Multiply

6

$6$ by

- 1

$- 1$ .

\frac{d}{d x} [x^{- 6}]

$\frac{d}{d x} [x^{- 6}]$

\frac{d}{d x} [x^{- 6}]

$\frac{d}{d x} [x^{- 6}]$

\frac{d}{d x} [x^{- 6}]

$\frac{d}{d x} [x^{- 6}]$

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 = - 6

$n = - 6$ .

- 6 x^{- 7}

$- 6 x^{- 7}$

Step 3

Simplify.

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Step 3.1

Rewrite the expression using the negative exponent rule

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

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

- 6 \frac{1}{x^{7}}

$- 6 \frac{1}{x^{7}}$

Step 3.2

Combine terms.

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Step 3.2.1

Combine

- 6

$- 6$ and

\frac{1}{x^{7}}

$\frac{1}{x^{7}}$ .

\frac{- 6}{x^{7}}

$\frac{- 6}{x^{7}}$

Step 3.2.2

Move the negative in front of the fraction.

- \frac{6}{x^{7}}

$- \frac{6}{x^{7}}$

- \frac{6}{x^{7}}

$- \frac{6}{x^{7}}$

- \frac{6}{x^{7}}

$- \frac{6}{x^{7}}$