Calculus Examples

y = \frac{1}{x}

$y = \frac{1}{x}$

Step 1

Differentiate both sides of the equation.

\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{x})

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1}{x})$

Step 2

The derivative of

y

$y$ with respect to

x

$x$ is

y'

$y'$ .

y'

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Rewrite

\frac{1}{x}

$\frac{1}{x}$ as

x^{- 1}

$x^{- 1}$ .

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

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

Step 3.2

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

$n = - 1$ .

- x^{- 2}

$- x^{- 2}$

Step 3.3

Rewrite the expression using the negative exponent rule

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

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

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

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

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

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

Step 4

Reform the equation by setting the left side equal to the right side.

y' = - \frac{1}{x^{2}}

$y' = - \frac{1}{x^{2}}$

Step 5

Replace

y'

$y'$ with

\frac{d y}{d x}

$\frac{d y}{d x}$ .

\frac{d y}{d x} = - \frac{1}{x^{2}}

$\frac{d y}{d x} = - \frac{1}{x^{2}}$