Calculus Examples

$f (x) = - \frac{1}{x}$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = - \frac{1}{x + h}$

Step 2.1.2

The final answer is $- \frac{1}{x + h}$ .

$- \frac{1}{x + h}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{1}{x + h}$

$f (x) = - \frac{1}{x}$

$f (x + h) = - \frac{1}{x + h}$

$f (x) = - \frac{1}{x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{1}{x + h} - (- \frac{1}{x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- - \frac{1}{x}$ .

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Step 4.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{1}{x + h} + 1 \frac{1}{x}}{h}$

Step 4.1.1.2

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

$\frac{- \frac{1}{x + h} + \frac{1}{x}}{h}$

Step 4.1.2

To write $- \frac{1}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{- \frac{1}{x + h} \cdot \frac{x}{x} + \frac{1}{x}}{h}$

Step 4.1.3

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

$\frac{- \frac{1}{x + h} \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 4.1.4

Write each expression with a common denominator of $(x + h) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.4.1

Multiply $\frac{1}{x + h}$ by $\frac{x}{x}$ .

$\frac{- \frac{x}{(x + h) x} + \frac{1}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 4.1.4.2

Multiply $\frac{1}{x}$ by $\frac{x + h}{x + h}$ .

$\frac{- \frac{x}{(x + h) x} + \frac{x + h}{x (x + h)}}{h}$

Step 4.1.4.3

Reorder the factors of $(x + h) x$ .

$\frac{- \frac{x}{x (x + h)} + \frac{x + h}{x (x + h)}}{h}$

Step 4.1.5

Combine the numerators over the common denominator.

$\frac{\frac{- x + x + h}{x (x + h)}}{h}$

Step 4.1.6

Rewrite $\frac{- x + x + h}{x (x + h)}$ in a factored form.

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Step 4.1.6.1

Add $- x$ and $x$ .

$\frac{\frac{0 + h}{x (x + h)}}{h}$

Step 4.1.6.2

Add $0$ and $h$ .

$\frac{\frac{h}{x (x + h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{h}{x (x + h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

$\frac{h}{x (x + h)} \cdot \frac{1}{h}$

Step 4.3.2

Rewrite the expression.

$\frac{1}{x (x + h)}$

Step 5