Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + h$ and $b = 1$ .

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

Step 2.1.2.3

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

$\sqrt{(x + h + 1) (x + h - 1)}$

Step 2.2

Find the components of the definition.

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

$f (x) = \sqrt{(x + 1) (x - 1)}$

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

$f (x) = \sqrt{(x + 1) (x - 1)}$

Step 3

Plug in the components.

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

Step 4

Multiply $- 1$ by $\sqrt{(x + 1) (x - 1)}$ .

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

Step 5