Calculus Examples

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

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

The final answer is $\sqrt{2 x + 2 h + 1}$ .

$\sqrt{2 x + 2 h + 1}$

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

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

$\frac{\sqrt{2 x + 2 h + 1} - \sqrt{2 x + 1}}{h}$

Step 5