Calculus Examples

$f (x) = x^{2} - \frac{16}{x}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{2} - \frac{16}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{16}{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{16}{x}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- \frac{16}{x}]$

Step 2

Evaluate $\frac{d}{d x} [- \frac{16}{x}]$ .

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Step 2.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- \frac{16}{x}$ with respect to $x$ is $- 16 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x - 16 \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

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

$2 x - 16 \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$2 x - 16 (- x^{- 2})$

Step 2.4

Multiply $- 1$ by $- 16$ .

$2 x + 16 x^{- 2}$

Step 3

Simplify.

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Step 3.1

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

$2 x + 16 \frac{1}{x^{2}}$

Step 3.2

Combine $16$ and $\frac{1}{x^{2}}$ .

$2 x + \frac{16}{x^{2}}$