Calculus Examples

$f (x) = \sqrt{x} - 5$

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

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

Step 2.1.2.2

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

$\sqrt{x + h} - 5$

Step 2.2

Find the components of the definition.

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

$f (x) = \sqrt{x} - 5$

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

$f (x) = \sqrt{x} - 5$

Step 3

Plug in the components.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 1$ by $- 5$ .

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

Step 4.3

Add $- 5$ and $5$ .

$\frac{\sqrt{x + h} + 0 - \sqrt{x}}{h}$

Step 4.4

Add $\sqrt{x + h}$ and $0$ .

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

Step 5