Calculus Examples

$f (x) = \frac{- 2}{x^{4}}$

Step 1

Move the negative in front of the fraction.

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

Step 2

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{4}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

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

Step 3

Apply basic rules of exponents.

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Step 3.1

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

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

Step 3.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2

Multiply $4$ by $- 1$ .

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

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$- 2 (- 4 x^{- 5})$

Step 5

Multiply $- 4$ by $- 2$ .

$8 x^{- 5}$

Step 6

Simplify.

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Step 6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$8 \frac{1}{x^{5}}$

Step 6.2

Combine $8$ and $\frac{1}{x^{5}}$ .

$\frac{8}{x^{5}}$