Calculus Examples

$\frac{1}{8 x^{2}}$

Step 1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{1}{8 x^{2}}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

Step 2

Apply basic rules of exponents.

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Step 2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 2$ .

$\frac{1}{8} (- 2 x^{- 3})$

Step 4

Simplify terms.

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Step 4.1

Combine $- 2$ and $\frac{1}{8}$ .

$\frac{- 2}{8} x^{- 3}$

Step 4.2

Combine $\frac{- 2}{8}$ and $x^{- 3}$ .

$\frac{- 2 x^{- 3}}{8}$

Step 4.3

Move $x^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 2}{8 x^{3}}$

Step 4.4

Cancel the common factor of $- 2$ and $8$ .

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Step 4.4.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{8 x^{3}}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $8 x^{3}$ .

$\frac{2 (- 1)}{2 (4 x^{3})}$

Step 4.4.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 (4 x^{3})}$

Step 4.4.2.3

Rewrite the expression.

$\frac{- 1}{4 x^{3}}$

Step 4.5

Move the negative in front of the fraction.

$- \frac{1}{4 x^{3}}$