Calculus Examples

$f (x) = \frac{1}{x}$ , $x = - 1$

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

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

$- x^{- 2}$

Step 1.3

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

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

Step 2

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{1}{{(- 1)}^{2}}$

Step 3

Raise $- 1$ to the power of $2$ .

$f' (- 1) = - \frac{1}{1}$

Step 4

Cancel the common factor of $1$ .

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Step 4.1

Cancel the common factor.

$f' (- 1) = - \frac{1}{1}$

Step 4.2

Rewrite the expression.

$f' (- 1) = - 1 \cdot 1$

Step 5

Multiply $- 1$ by $1$ .

$f' (- 1) = - 1$