Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

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

$\sqrt{x + h + 3}$

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

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

$\frac{\sqrt{x + h + 3} - \sqrt{x + 3}}{h}$

Step 5