Precalculus Examples

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

Step 1

Evaluate the numerator in $\frac{f (x + h) - f (x)}{h}$ .

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Step 1.1

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

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

Step 1.2

Replace the function designators in $f (x + h) - f (x)$ with the actual functions.

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

Step 1.3

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

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Step 1.3.1

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

$\frac{3}{x + h - 1} - \frac{3}{x - 1}$

Step 1.3.2

To write $\frac{3}{x + h - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 1.3.3

To write $- \frac{3}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + h - 1}{x + h - 1}$ .

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

Step 1.3.4

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

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Step 1.3.4.1

Multiply $\frac{3}{x + h - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 1.3.4.2

Multiply $\frac{3}{x - 1}$ by $\frac{x + h - 1}{x + h - 1}$ .

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

Step 1.3.4.3

Reorder the factors of $(x - 1) (x + h - 1)$ .

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

Step 1.3.5

Combine the numerators over the common denominator.

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

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

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

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Step 1.3.6.1.1

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

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

Step 1.3.6.1.2

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

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

Step 1.3.6.2

Apply the distributive property.

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

Step 1.3.6.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.6.4

Subtract $x$ from $x$ .

$\frac{3 (0 - 1 - h + 1)}{(x + h - 1) (x - 1)}$

Step 1.3.6.5

Subtract $1$ from $0$ .

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

Step 1.3.6.6

Add $- 1$ and $1$ .

$\frac{3 (- h + 0)}{(x + h - 1) (x - 1)}$

Step 1.3.6.7

Add $- h$ and $0$ .

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

Step 1.3.6.8

Combine exponents.

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Step 1.3.6.8.1

Factor out negative.

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

Step 1.3.6.8.2

Multiply $3$ by $- 1$ .

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

Step 1.3.7

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Cancel the common factor of $h$ .

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Step 3.2.1

Move the leading negative in $- \frac{3 h}{(x + h - 1) (x - 1)}$ into the numerator.

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Move the negative in front of the fraction.

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