Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Move $- 1$ to the left of $n$ .

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

Step 1.2.3

Rewrite $- 1 n$ as $- n$ .

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

Step 2

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

$- n x^{- n - 1}$