Calculus Examples

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

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [\frac{x}{2}]$ .

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Step 2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{2}{x}]$

Step 2.2

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

$\frac{1}{2} \cdot 1 + \frac{d}{d x} [\frac{2}{x}]$

Step 2.3

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} + \frac{d}{d x} [\frac{2}{x}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{2}{x}]$ .

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Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{2} + 2 \frac{d}{d x} [\frac{1}{x}]$

Step 3.2

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

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

Step 3.3

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

$\frac{1}{2} + 2 (- x^{- 2})$

Step 3.4

Multiply $- 1$ by $2$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Combine terms.

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Step 5.1.1

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

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

Step 5.1.2

Move the negative in front of the fraction.

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

Step 5.2

Reorder terms.

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