Precalculus Examples

$g (x) = \frac{1}{3} \sqrt[3]{x + 27}$

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.

$g (x + h) = \frac{1}{3} \cdot \sqrt[3]{(x + h) + 27}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Combine $\frac{1}{3}$ and $\sqrt[3]{x + h + 27}$ .

$g (x + h) = \frac{\sqrt[3]{x + h + 27}}{3}$

Step 2.1.2.2

The final answer is $\frac{\sqrt[3]{x + h + 27}}{3}$ .

$\frac{\sqrt[3]{x + h + 27}}{3}$

Step 2.2

Find the components of the definition.

$g (x + h) = \frac{\sqrt[3]{x + h + 27}}{3}$

$g (x) = \frac{\sqrt[3]{x + 27}}{3}$

$g (x + h) = \frac{\sqrt[3]{x + h + 27}}{3}$

$g (x) = \frac{\sqrt[3]{x + 27}}{3}$

Step 3

Plug in the components.

$\frac{g (x + h) - g x}{h} = \frac{\frac{\sqrt[3]{x + h + 27}}{3} - (\frac{\sqrt[3]{x + 27}}{3})}{h}$

Step 4

Simplify.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt[3]{x + h + 27} - \sqrt[3]{x + 27}}{3}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt[3]{x + h + 27} - \sqrt[3]{x + 27}}{3} \cdot \frac{1}{h}$

Step 4.3

Multiply $\frac{\sqrt[3]{x + h + 27} - \sqrt[3]{x + 27}}{3}$ by $\frac{1}{h}$ .

$\frac{\sqrt[3]{x + h + 27} - \sqrt[3]{x + 27}}{3 h}$

Step 5