Calculus Examples

$\frac{1}{2 x}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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