Precalculus Examples

$f (x) = 10 \sqrt[3]{x} - 5$

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.

$f (x + h) = 10 \sqrt[3]{x + h} - 5$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = 10 \sqrt[3]{x + h} - 5$

Step 2.1.2.2

The final answer is $10 \sqrt[3]{x + h} - 5$ .

$10 \sqrt[3]{x + h} - 5$

Step 2.2

Find the components of the definition.

$f (x + h) = 10 \sqrt[3]{x + h} - 5$

$f (x) = 10 \sqrt[3]{x} - 5$

$f (x + h) = 10 \sqrt[3]{x + h} - 5$

$f (x) = 10 \sqrt[3]{x} - 5$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{10 \sqrt[3]{x + h} - 5 - (10 \sqrt[3]{x} - 5)}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + h}$ as ${(x + h)}^{\frac{1}{3}}$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 5 - (10 \sqrt[3]{x} - 5)}{h}$

Step 4.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 5 - (10 x^{\frac{1}{3}} - 5)}{h}$

Step 4.3

Apply the distributive property.

$\frac{10 {(x + h)}^{\frac{1}{3}} - 5 - (10 x^{\frac{1}{3}}) - - 5}{h}$

Step 4.4

Multiply $10$ by $- 1$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 5 - 10 x^{\frac{1}{3}} - - 5}{h}$

Step 4.5

Multiply $- 1$ by $- 5$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 5 - 10 x^{\frac{1}{3}} + 5}{h}$

Step 4.6

Add $- 5$ and $5$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 10 x^{\frac{1}{3}} + 0}{h}$

Step 4.7

Add $10 {(x + h)}^{\frac{1}{3}} - 10 x^{\frac{1}{3}}$ and $0$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 10 x^{\frac{1}{3}}}{h}$

Step 4.8

Factor $10$ out of $10 {(x + h)}^{\frac{1}{3}} - 10 x^{\frac{1}{3}}$ .

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Step 4.8.1

Factor $10$ out of $10 {(x + h)}^{\frac{1}{3}}$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} - 10 x^{\frac{1}{3}}}{h}$

Step 4.8.2

Factor $10$ out of $- 10 x^{\frac{1}{3}}$ .

$\frac{10 {(x + h)}^{\frac{1}{3}} + 10 (- x^{\frac{1}{3}})}{h}$

Step 4.8.3

Factor $10$ out of $10 {(x + h)}^{\frac{1}{3}} + 10 (- x^{\frac{1}{3}})$ .

$\frac{10 ({(x + h)}^{\frac{1}{3}} - x^{\frac{1}{3}})}{h}$

Step 5