Calculus Examples

$\sqrt[3]{x^{2}}$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{d}{d x} [x^{\frac{2}{3}}]$

Step 3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} - 1}$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}$

Step 5

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}$

Step 6

Combine the numerators over the common denominator.

$\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

$\frac{2}{3} x^{\frac{2 - 3}{3}}$

Step 7.2

Subtract $3$ from $2$ .

$\frac{2}{3} x^{\frac{- 1}{3}}$

Step 8

Move the negative in front of the fraction.

$\frac{2}{3} x^{- \frac{1}{3}}$

Step 9

Simplify.

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Step 9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}}$

Step 9.2

Multiply $\frac{2}{3}$ by $\frac{1}{x^{\frac{1}{3}}}$ .

$\frac{2}{3 x^{\frac{1}{3}}}$