Precalculus Examples

$f (x) = x - 4 \sqrt{x}$

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.

$f (x + h) = (x + h) - 4 \sqrt{x + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = x + h - 4 \sqrt{x + h}$

Step 2.1.2.2

The final answer is $x + h - 4 \sqrt{x + h}$ .

$x + h - 4 \sqrt{x + h}$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $x$ and $h$ .

$x + h - 4 \sqrt{h + x}$

Step 2.2.2

Reorder $x$ and $h$ .

$h + x - 4 \sqrt{h + x}$

Step 2.3

Find the components of the definition.

$f (x + h) = h + x - 4 \sqrt{h + x}$

$f (x) = x - 4 \sqrt{x}$

$f (x + h) = h + x - 4 \sqrt{h + x}$

$f (x) = x - 4 \sqrt{x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h + x - 4 \sqrt{h + x} - (x - 4 \sqrt{x})}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{h + x - 4 \sqrt{h + x} - x - (- 4 \sqrt{x})}{h}$

Step 4.2

Multiply $- 4$ by $- 1$ .

$\frac{h + x - 4 \sqrt{h + x} - x + 4 \sqrt{x}}{h}$

Step 4.3

Subtract $x$ from $x$ .

$\frac{h + 0 - 4 \sqrt{h + x} + 4 \sqrt{x}}{h}$

Step 4.4

Add $h$ and $0$ .

$\frac{h - 4 \sqrt{h + x} + 4 \sqrt{x}}{h}$

Step 5