Precalculus Examples

$f (x) = \frac{3 - 2 x}{x}$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

The final answer is $\frac{3 - 2 x - 2 h}{x + h}$ .

$\frac{3 - 2 x - 2 h}{x + h}$

Step 2.2

Find the components of the definition.

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

$f (x) = \frac{3 - 2 x}{x}$

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

$f (x) = \frac{3 - 2 x}{x}$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{3 - 2 x - 2 h}{x + h}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 4.1.2

To write $- \frac{3 - 2 x}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

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

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 $\frac{3 - 2 x - 2 h}{x + h}$ by $\frac{x}{x}$ .

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

Step 4.1.3.2

Multiply $\frac{3 - 2 x}{x}$ by $\frac{x + h}{x + h}$ .

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Rewrite $\frac{(3 - 2 x - 2 h) x - (3 - 2 x) (x + h)}{x (x + h)}$ in a factored form.

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Step 4.1.5.1

Apply the distributive property.

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

Step 4.1.5.2

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

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Step 4.1.5.2.1

Move $x$ .

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

Step 4.1.5.2.2

Multiply $x$ by $x$ .

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

Step 4.1.5.3

Apply the distributive property.

$\frac{\frac{3 x - 2 x^{2} - 2 h x + (- 1 \cdot 3 - (- 2 x)) (x + h)}{x (x + h)}}{h}$

Step 4.1.5.4

Multiply $- 1$ by $3$ .

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

Step 4.1.5.5

Multiply $- 2$ by $- 1$ .

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

Step 4.1.5.6

Expand $(- 3 + 2 x) (x + h)$ using the FOIL Method.

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Step 4.1.5.6.1

Apply the distributive property.

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

Step 4.1.5.6.2

Apply the distributive property.

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

Step 4.1.5.6.3

Apply the distributive property.

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

Step 4.1.5.7

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

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Step 4.1.5.7.1

Move $x$ .

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

Step 4.1.5.7.2

Multiply $x$ by $x$ .

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

Step 4.1.5.8

Subtract $3 x$ from $3 x$ .

$\frac{\frac{- 2 x^{2} - 2 h x + 0 - 3 h + 2 x^{2} + 2 x h}{x (x + h)}}{h}$

Step 4.1.5.9

Add $- 2 x^{2}$ and $0$ .

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

Step 4.1.5.10

Add $- 2 x^{2}$ and $2 x^{2}$ .

$\frac{\frac{- 2 h x - 3 h + 0 + 2 x h}{x (x + h)}}{h}$

Step 4.1.5.11

Add $- 2 h x$ and $0$ .

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

Step 4.1.5.12

Add $- 2 h x$ and $2 x h$ .

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Step 4.1.5.12.1

Move $h$ .

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

Step 4.1.5.12.2

Add $- 2 x h$ and $2 x h$ .

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

Step 4.1.5.13

Add $- 3 h$ and $0$ .

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

Step 4.1.6

Move the negative in front of the fraction.

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

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

Move the leading negative in $- \frac{3 h}{x (x + h)}$ into the numerator.

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

Step 4.3.2

Factor $h$ out of $- 3 h$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5