Calculus Examples

- \frac{1}{x}

$- \frac{1}{x}$

Step 1

Differentiate using the Product Rule which states that

\frac{d}{d x} [f (x) g (x)]

$\frac{d}{d x} [f (x) g (x)]$ is

f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]

$f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where

f (x) = - 1

$f (x) = - 1$ and

g (x) = \frac{1}{x}

$g (x) = \frac{1}{x}$ .

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

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

Step 2

Differentiate.

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Step 2.1

Rewrite

\frac{1}{x}

$\frac{1}{x}$ as

x^{- 1}

$x^{- 1}$ .

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

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

Step 2.2

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

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

Step 2.3

Multiply.

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Step 2.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 2.3.2

Multiply

x^{- 2}

$x^{- 2}$ by

1

$1$ .

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

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

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

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

Step 2.4

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- 1

$- 1$ with respect to

x

$x$ is

0

$0$ .

x^{- 2} + \frac{1}{x} \cdot 0

$x^{- 2} + \frac{1}{x} \cdot 0$

Step 2.5

Simplify the expression.

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Step 2.5.1

Multiply

\frac{1}{x}

$\frac{1}{x}$ by

0

$0$ .

x^{- 2} + 0

$x^{- 2} + 0$

Step 2.5.2

Add

x^{- 2}

$x^{- 2}$ and

0

$0$ .

x^{- 2}

$x^{- 2}$

x^{- 2}

$x^{- 2}$

x^{- 2}

$x^{- 2}$

Step 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}}

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