Precalculus Examples

$f (x) = \frac{2 x}{x - 1}$

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.

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

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

The final answer is $\frac{2 (x + h)}{x + h - 1}$ .

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

Step 2.2

Find the components of the definition.

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

$f (x) = \frac{2 x}{x - 1}$

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

$f (x) = \frac{2 x}{x - 1}$

Step 3

Plug in the components.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

To write $\frac{2 (x + h)}{x + h - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4.1.2

To write $- \frac{2 x}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + h - 1}{x + h - 1}$ .

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

Step 4.1.3

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

Tap for more steps...

Step 4.1.3.1

Multiply $\frac{2 (x + h)}{x + h - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 4.1.3.2

Multiply $\frac{2 x}{x - 1}$ by $\frac{x + h - 1}{x + h - 1}$ .

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Rewrite $\frac{2 (x + h) (x - 1) - 2 x (x + h - 1)}{(x + h - 1) (x - 1)}$ in a factored form.

Tap for more steps...

Step 4.1.5.1

Factor $2$ out of $2 (x + h) (x - 1) - 2 x (x + h - 1)$ .

Tap for more steps...

Step 4.1.5.1.1

Factor $2$ out of $2 (x + h) (x - 1)$ .

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

Step 4.1.5.1.2

Factor $2$ out of $- 2 x (x + h - 1)$ .

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

Step 4.1.5.1.3

Factor $2$ out of $2 ((x + h) (x - 1)) + 2 (- x (x + h - 1))$ .

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

Step 4.1.5.2

Expand $(x + h) (x - 1)$ using the FOIL Method.

Tap for more steps...

Step 4.1.5.2.1

Apply the distributive property.

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

Step 4.1.5.2.2

Apply the distributive property.

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

Step 4.1.5.2.3

Apply the distributive property.

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

Step 4.1.5.3

Simplify each term.

Tap for more steps...

Step 4.1.5.3.1

Multiply $x$ by $x$ .

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

Step 4.1.5.3.2

Move $- 1$ to the left of $x$ .

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

Step 4.1.5.3.3

Rewrite $- 1 x$ as $- x$ .

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

Step 4.1.5.3.4

Move $- 1$ to the left of $h$ .

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

Step 4.1.5.3.5

Rewrite $- 1 h$ as $- h$ .

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

Step 4.1.5.4

Apply the distributive property.

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

Step 4.1.5.5

Simplify.

Tap for more steps...

Step 4.1.5.5.1

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

Tap for more steps...

Step 4.1.5.5.1.1

Move $x$ .

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

Step 4.1.5.5.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.5.2

Multiply $- x \cdot - 1$ .

Tap for more steps...

Step 4.1.5.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5.5.2.2

Multiply $x$ by $1$ .

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

Step 4.1.5.6

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

$\frac{\frac{2 (- x + h x - h + 0 - x h + x)}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.7

Add $- x$ and $0$ .

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

Step 4.1.5.8

Add $- x$ and $x$ .

$\frac{\frac{2 (h x - h - x h + 0)}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.9

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

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

Step 4.1.5.10

Subtract $x h$ from $h x$ .

Tap for more steps...

Step 4.1.5.10.1

Reorder $h$ and $x$ .

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

Step 4.1.5.10.2

Subtract $x h$ from $x h$ .

$\frac{\frac{2 (- h + 0)}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.11

Add $- h$ and $0$ .

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

Step 4.1.5.12

Combine exponents.

Tap for more steps...

Step 4.1.5.12.1

Factor out negative.

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

Step 4.1.5.12.2

Multiply $2$ by $- 1$ .

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

Step 4.1.6

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.3.1

Move the leading negative in $- \frac{2 h}{(x + h - 1) (x - 1)}$ into the numerator.

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

Step 4.3.2

Factor $h$ out of $- 2 h$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5