Calculus Examples

$f (x) = - \frac{10}{x}$ $x = - 12$

Step 1

Find the derivative.

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Step 1.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- \frac{10}{x}$ with respect to $x$ is $- 10 \frac{d}{d x} [\frac{1}{x}]$ .

$- 10 \frac{d}{d x} [\frac{1}{x}]$

Step 1.2

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

$- 10 \frac{d}{d x} [x^{- 1}]$

Step 1.3

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

$- 10 (- x^{- 2})$

Step 1.4

Multiply $- 1$ by $- 10$ .

$10 x^{- 2}$

Step 1.5

Simplify.

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Step 1.5.1

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

$10 \frac{1}{x^{2}}$

Step 1.5.2

Combine $10$ and $\frac{1}{x^{2}}$ .

$\frac{10}{x^{2}}$

Step 2

Replace the variable $x$ with $- 12$ in the expression.

$f' (- 12) = \frac{10}{{(- 12)}^{2}}$

Step 3

Raise $- 12$ to the power of $2$ .

$f' (- 12) = \frac{10}{144}$

Step 4

Cancel the common factor of $10$ and $144$ .

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Step 4.1

Factor $2$ out of $10$ .

$f' (- 12) = \frac{2 (5)}{144}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $2$ out of $144$ .

$f' (- 12) = \frac{2 \cdot 5}{2 \cdot 72}$

Step 4.2.2

Cancel the common factor.

$f' (- 12) = \frac{2 \cdot 5}{2 \cdot 72}$

Step 4.2.3

Rewrite the expression.

$f' (- 12) = \frac{5}{72}$