Find the Difference Quotient f(x) = square root of 2x+6

$f (x) = \sqrt{2 x + 6}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

$f (x + h) = \sqrt{2 (x + h) + 6}$

Simplify the result.

Factor $2$ out of $2 (x + h) + 6$ .

Factor $2$ out of $6$ .

$f (x + h) = \sqrt{2 (x + h) + 2 \cdot 3}$

Factor $2$ out of $2 (x + h) + 2 \cdot 3$ .

$f (x + h) = \sqrt{2 (x + h + 3)}$

The final answer is $\sqrt{2 (x + h + 3)}$ .

$\sqrt{2 (x + h + 3)}$

Find the components of the definition.

$f (x + h) = \sqrt{2 (x + h + 3)}$

$f (x) = \sqrt{2 (x + 3)}$

$f (x + h) = \sqrt{2 (x + h + 3)}$

$f (x) = \sqrt{2 (x + 3)}$

Step 3

Plug in the components.

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

Step 4

Multiply $- 1$ by $\sqrt{2 (x + 3)}$ .

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

Step 5