Precalculus Examples

$\frac{1}{x}$

Step 1

Write $\frac{1}{x}$ as a function.

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

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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 5.1.2

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 5.1.3

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 5.1.3.1

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 5.1.3.2

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

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

Step 5.1.3.3

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

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

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

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Step 5.1.5.1

Apply the distributive property.

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

Step 5.1.5.2

Subtract $x$ from $x$ .

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

Step 5.1.5.3

Subtract $h$ from $0$ .

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

Step 5.1.6

Move the negative in front of the fraction.

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3

Cancel the common factor of $h$ .

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Step 5.3.1

Move the leading negative in $- \frac{h}{x (x + h)}$ into the numerator.

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

Step 5.3.2

Factor $h$ out of $- h$ .

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

Step 5.3.3

Cancel the common factor.

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

Step 5.3.4

Rewrite the expression.

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

Step 5.4

Move the negative in front of the fraction.

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

Step 6