Calculus Examples

\frac{1}{2 x}

$\frac{1}{2 x}$

Step 1

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

x

$x$ , the derivative of

\frac{1}{2 x}

$\frac{1}{2 x}$ with respect to

x

$x$ is

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

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

\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}

$\frac{1}{x}$ as

x^{- 1}

$x^{- 1}$ .

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

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = - 1

$n = - 1$ .

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

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

Step 4

Combine fractions.

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Step 4.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

x^{- 2}

$x^{- 2}$ .

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

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

Step 4.2

Move

x^{- 2}

$x^{- 2}$ to the denominator using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

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

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