Precalculus Examples

$f (x) = - \frac{5}{x}$ $x = 11$

Step 1

Find the derivative.

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Step 1.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- \frac{5}{x}$ with respect to $x$ is $- 5 \frac{d}{d x} [\frac{1}{x}]$ .

$- 5 \frac{d}{d x} [\frac{1}{x}]$

Step 1.2

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

$- 5 \frac{d}{d x} [x^{- 1}]$

Step 1.3

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

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

Step 1.4

Multiply $- 1$ by $- 5$ .

$5 x^{- 2}$

Step 1.5

Simplify.

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Step 1.5.1

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

$5 \frac{1}{x^{2}}$

Step 1.5.2

Combine $5$ and $\frac{1}{x^{2}}$ .

$\frac{5}{x^{2}}$

Step 2

Replace the variable $x$ with $11$ in the expression.

$f' (11) = \frac{5}{{(11)}^{2}}$

Step 3

Raise $11$ to the power of $2$ .

$f' (11) = \frac{5}{121}$