Calculus Examples

\frac{8}{x^{2}}

$\frac{8}{x^{2}}$

Step 1

Since

8

$8$ is constant with respect to

x

$x$ , the derivative of

\frac{8}{x^{2}}

$\frac{8}{x^{2}}$ with respect to

x

$x$ is

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

$8 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

$8 \frac{d}{d x} [\frac{1}{x^{2}}]$

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}

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

$8 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 2.2

Multiply the exponents in

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

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Step 2.2.1

Apply the power rule and multiply exponents,

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

$8 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 2.2.2

Multiply

$2$ by

$- 1$ .

$8 \frac{d}{d x} [x^{- 2}]$

Step 3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = - 2$ .

$8 (- 2 x^{- 3})$

Step 4

Multiply

$- 2$ by

$8$ .

$- 16 x^{- 3}$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule

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

$- 16 \frac{1}{x^{3}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine

$- 16$ and

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

$\frac{- 16}{x^{3}}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{16}{x^{3}}$

$\frac{8}{x^{2}}$

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