Precalculus Examples

$f (x) = - \frac{3 x}{2} + 4$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$f (x + h) = - \frac{3 (x + h)}{2} + 4$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (x + h) = - \frac{3 (x + h)}{2} + 4 \cdot \frac{2}{2}$

Step 2.1.2.2

Simplify terms.

Tap for more steps...

Step 2.1.2.2.1

Combine $4$ and $\frac{2}{2}$ .

$f (x + h) = - \frac{3 (x + h)}{2} + \frac{4 \cdot 2}{2}$

Step 2.1.2.2.2

Combine the numerators over the common denominator.

$f (x + h) = \frac{- 3 (x + h) + 4 \cdot 2}{2}$

Step 2.1.2.3

Simplify the numerator.

Tap for more steps...

Step 2.1.2.3.1

Apply the distributive property.

$f (x + h) = \frac{- 3 x - 3 h + 4 \cdot 2}{2}$

Step 2.1.2.3.2

Multiply $4$ by $2$ .

$f (x + h) = \frac{- 3 x - 3 h + 8}{2}$

Step 2.1.2.4

Simplify with factoring out.

Tap for more steps...

Step 2.1.2.4.1

Factor $- 1$ out of $- 3 x$ .

$f (x + h) = \frac{- (3 x) - 3 h + 8}{2}$

Step 2.1.2.4.2

Factor $- 1$ out of $- 3 h$ .

$f (x + h) = \frac{- (3 x) - (3 h) + 8}{2}$

Step 2.1.2.4.3

Factor $- 1$ out of $- (3 x) - (3 h)$ .

$f (x + h) = \frac{- (3 x + 3 h) + 8}{2}$

Step 2.1.2.4.4

Rewrite $8$ as $- 1 (- 8)$ .

$f (x + h) = \frac{- (3 x + 3 h) - 1 \cdot - 8}{2}$

Step 2.1.2.4.5

Factor $- 1$ out of $- (3 x + 3 h) - 1 (- 8)$ .

$f (x + h) = \frac{- (3 x + 3 h - 8)}{2}$

Step 2.1.2.4.6

Simplify the expression.

Tap for more steps...

Step 2.1.2.4.6.1

Rewrite $- (3 x + 3 h - 8)$ as $- 1 (3 x + 3 h - 8)$ .

$f (x + h) = \frac{- 1 (3 x + 3 h - 8)}{2}$

Step 2.1.2.4.6.2

Move the negative in front of the fraction.

$f (x + h) = - \frac{3 x + 3 h - 8}{2}$

Step 2.1.2.5

The final answer is $- \frac{3 x + 3 h - 8}{2}$ .

$- \frac{3 x + 3 h - 8}{2}$

Step 2.2

Find the components of the definition.

$f (x + h) = - \frac{3 x + 3 h - 8}{2}$

$f (x) = - \frac{3 x}{2} + 4$

$f (x + h) = - \frac{3 x + 3 h - 8}{2}$

$f (x) = - \frac{3 x}{2} + 4$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{3 x + 3 h - 8}{2} - (- \frac{3 x}{2} + 4)}{h}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$\frac{- \frac{3 x + 3 h - 8}{2} - - \frac{3 x}{2} - 1 \cdot 4}{h}$

Step 4.1.2

Multiply $- - \frac{3 x}{2}$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{3 x + 3 h - 8}{2} + 1 \frac{3 x}{2} - 1 \cdot 4}{h}$

Step 4.1.2.2

Multiply $\frac{3 x}{2}$ by $1$ .

$\frac{- \frac{3 x + 3 h - 8}{2} + \frac{3 x}{2} - 1 \cdot 4}{h}$

Step 4.1.3

Multiply $- 1$ by $4$ .

$\frac{- \frac{3 x + 3 h - 8}{2} + \frac{3 x}{2} - 4}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{- (3 x + 3 h - 8) + 3 x}{2} - 4}{h}$

Step 4.1.5

Simplify the numerator.

Tap for more steps...

Step 4.1.5.1

Apply the distributive property.

$\frac{\frac{- (3 x) - (3 h) - - 8 + 3 x}{2} - 4}{h}$

Step 4.1.5.2

Simplify.

Tap for more steps...

Step 4.1.5.2.1

Multiply $3$ by $- 1$ .

$\frac{\frac{- 3 x - (3 h) - - 8 + 3 x}{2} - 4}{h}$

Step 4.1.5.2.2

Multiply $3$ by $- 1$ .

$\frac{\frac{- 3 x - 3 h - - 8 + 3 x}{2} - 4}{h}$

Step 4.1.5.2.3

Multiply $- 1$ by $- 8$ .

$\frac{\frac{- 3 x - 3 h + 8 + 3 x}{2} - 4}{h}$

Step 4.1.5.3

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

$\frac{\frac{- 3 h + 8 + 0}{2} - 4}{h}$

Step 4.1.5.4

Add $- 3 h + 8$ and $0$ .

$\frac{\frac{- 3 h + 8}{2} - 4}{h}$

Step 4.1.6

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{- 3 h + 8}{2} - 4 \cdot \frac{2}{2}}{h}$

Step 4.1.7

Combine $- 4$ and $\frac{2}{2}$ .

$\frac{\frac{- 3 h + 8}{2} + \frac{- 4 \cdot 2}{2}}{h}$

Step 4.1.8

Combine the numerators over the common denominator.

$\frac{\frac{- 3 h + 8 - 4 \cdot 2}{2}}{h}$

Step 4.1.9

Simplify the numerator.

Tap for more steps...

Step 4.1.9.1

Multiply $- 4$ by $2$ .

$\frac{\frac{- 3 h + 8 - 8}{2}}{h}$

Step 4.1.9.2

Subtract $8$ from $8$ .

$\frac{\frac{- 3 h + 0}{2}}{h}$

Step 4.1.9.3

Add $- 3 h$ and $0$ .

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

Step 4.1.10

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{3 h}{2} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.3.1

Move the leading negative in $- \frac{3 h}{2}$ into the numerator.

$\frac{- 3 h}{2} \cdot \frac{1}{h}$

Step 4.3.2

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

$\frac{h \cdot - 3}{2} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 3}{2} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 3}{2}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{3}{2}$

Step 5