Calculus Examples

$f (x) = \sqrt{x^{2} + 5}$ , $x = 2$

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.

$f (x + h) = \sqrt{{(x + h)}^{2} + 5}$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = \sqrt{(x + h) (x + h) + 5}$

Step 2.1.2.2

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

Tap for more steps...

Step 2.1.2.2.1

Apply the distributive property.

$f (x + h) = \sqrt{x (x + h) + h (x + h) + 5}$

Step 2.1.2.2.2

Apply the distributive property.

$f (x + h) = \sqrt{x \cdot x + x h + h (x + h) + 5}$

Step 2.1.2.2.3

Apply the distributive property.

$f (x + h) = \sqrt{x \cdot x + x h + h x + h \cdot h + 5}$

Step 2.1.2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = \sqrt{x^{2} + x h + h x + h \cdot h + 5}$

Step 2.1.2.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = \sqrt{x^{2} + x h + h x + h^{2} + 5}$

Step 2.1.2.3.2

Add $x h$ and $h x$ .

Tap for more steps...

Step 2.1.2.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = \sqrt{x^{2} + h x + h x + h^{2} + 5}$

Step 2.1.2.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = \sqrt{x^{2} + 2 h x + h^{2} + 5}$

Step 2.1.2.4

The final answer is $\sqrt{x^{2} + 2 h x + h^{2} + 5}$ .

$\sqrt{x^{2} + 2 h x + h^{2} + 5}$

Step 2.2

Find the components of the definition.

$f (x + h) = \sqrt{x^{2} + 2 h x + h^{2} + 5}$

$f (x) = \sqrt{x^{2} + 5}$

$f (x + h) = \sqrt{x^{2} + 2 h x + h^{2} + 5}$

$f (x) = \sqrt{x^{2} + 5}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\sqrt{x^{2} + 2 h x + h^{2} + 5} - (\sqrt{x^{2} + 5})}{h}$

Step 4

Multiply $- 1$ by $\sqrt{x^{2} + 5}$ .

$\frac{\sqrt{x^{2} + 2 h x + h^{2} + 5} - \sqrt{x^{2} + 5}}{h}$

Step 5

Replace the variable $x$ with $2$ in the expression.

$\frac{\sqrt{{(2)}^{2} + 2 h (2) + h^{2} + 5} - \sqrt{{(2)}^{2} + 5}}{h}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Raise $2$ to the power of $2$ .

$\frac{\sqrt{4 + 2 h \cdot 2 + h^{2} + 5} - \sqrt{2^{2} + 5}}{h}$

Step 6.2

Multiply $2$ by $2$ .

$\frac{\sqrt{4 + 4 h + h^{2} + 5} - \sqrt{2^{2} + 5}}{h}$

Step 6.3

Add $4$ and $5$ .

$\frac{\sqrt{4 h + h^{2} + 9} - \sqrt{2^{2} + 5}}{h}$

Step 6.4

Reorder terms.

$\frac{\sqrt{h^{2} + 4 h + 9} - \sqrt{2^{2} + 5}}{h}$

Step 6.5

Raise $2$ to the power of $2$ .

$\frac{\sqrt{h^{2} + 4 h + 9} - \sqrt{4 + 5}}{h}$

Step 6.6

Add $4$ and $5$ .

$\frac{\sqrt{h^{2} + 4 h + 9} - \sqrt{9}}{h}$

Step 6.7

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{h^{2} + 4 h + 9} - \sqrt{3^{2}}}{h}$

Step 6.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{h^{2} + 4 h + 9} - 1 \cdot 3}{h}$

Step 6.9

Multiply $- 1$ by $3$ .

$\frac{\sqrt{h^{2} + 4 h + 9} - 3}{h}$

Step 7