Precalculus Examples

$f (x) = \frac{x - 3}{x}$

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.

$f (x + h) = \frac{(x + h) - 3}{x + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{x + h - 3}{x + h}$

Step 2.1.2.2

The final answer is $\frac{x + h - 3}{x + h}$ .

$\frac{x + h - 3}{x + h}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{x + h - 3}{x + h}$

$f (x) = \frac{x - 3}{x}$

$f (x + h) = \frac{x + h - 3}{x + h}$

$f (x) = \frac{x - 3}{x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{x + h - 3}{x + h} - (\frac{x - 3}{x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{x + h - 3}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{\frac{x + h - 3}{x + h} \cdot \frac{x}{x} - \frac{x - 3}{x}}{h}$

Step 4.1.2

To write $- \frac{x - 3}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

$\frac{\frac{x + h - 3}{x + h} \cdot \frac{x}{x} - \frac{x - 3}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 4.1.3

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 4.1.3.1

Multiply $\frac{x + h - 3}{x + h}$ by $\frac{x}{x}$ .

$\frac{\frac{(x + h - 3) x}{(x + h) x} - \frac{x - 3}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 4.1.3.2

Multiply $\frac{x - 3}{x}$ by $\frac{x + h}{x + h}$ .

$\frac{\frac{(x + h - 3) x}{(x + h) x} - \frac{(x - 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.3.3

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

$\frac{\frac{(x + h - 3) x}{x (x + h)} - \frac{(x - 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{(x + h - 3) x - (x - 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.5

Rewrite $\frac{(x + h - 3) x - (x - 3) (x + h)}{x (x + h)}$ in a factored form.

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Step 4.1.5.1

Apply the distributive property.

$\frac{\frac{x \cdot x + h x - 3 x - (x - 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.5.2

Multiply $x$ by $x$ .

$\frac{\frac{x^{2} + h x - 3 x - (x - 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.5.3

Apply the distributive property.

$\frac{\frac{x^{2} + h x - 3 x + (- x - - 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.5.4

Multiply $- 1$ by $- 3$ .

$\frac{\frac{x^{2} + h x - 3 x + (- x + 3) (x + h)}{x (x + h)}}{h}$

Step 4.1.5.5

Expand $(- x + 3) (x + h)$ using the FOIL Method.

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Step 4.1.5.5.1

Apply the distributive property.

$\frac{\frac{x^{2} + h x - 3 x - x (x + h) + 3 (x + h)}{x (x + h)}}{h}$

Step 4.1.5.5.2

Apply the distributive property.

$\frac{\frac{x^{2} + h x - 3 x - x \cdot x - x h + 3 (x + h)}{x (x + h)}}{h}$

Step 4.1.5.5.3

Apply the distributive property.

$\frac{\frac{x^{2} + h x - 3 x - x \cdot x - x h + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.6

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.6.1

Move $x$ .

$\frac{\frac{x^{2} + h x - 3 x - (x \cdot x) - x h + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.6.2

Multiply $x$ by $x$ .

$\frac{\frac{x^{2} + h x - 3 x - x^{2} - x h + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.7

Subtract $x^{2}$ from $x^{2}$ .

$\frac{\frac{h x - 3 x + 0 - x h + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.8

Add $h x$ and $0$ .

$\frac{\frac{h x - 3 x - x h + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.9

Subtract $x h$ from $h x$ .

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Step 4.1.5.9.1

Reorder $h$ and $x$ .

$\frac{\frac{- 3 x + x h - x h + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.9.2

Subtract $x h$ from $x h$ .

$\frac{\frac{- 3 x + 0 + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.10

Add $- 3 x$ and $0$ .

$\frac{\frac{- 3 x + 3 x + 3 h}{x (x + h)}}{h}$

Step 4.1.5.11

Add $- 3 x$ and $3 x$ .

$\frac{\frac{0 + 3 h}{x (x + h)}}{h}$

Step 4.1.5.12

Add $0$ and $3 h$ .

$\frac{\frac{3 h}{x (x + h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 h}{x (x + h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $3 h$ .

$\frac{h \cdot 3}{x (x + h)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h \cdot 3}{x (x + h)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{3}{x (x + h)}$

Step 5