Calculus Examples

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

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

Step 1

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = \frac{2}{3}

$n = \frac{2}{3}$ .

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

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

Step 2

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 3

Combine

- 1

$- 1$ and

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 5.2

Subtract

3

$3$ from

2

$2$ .

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

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

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

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

Step 6

Move the negative in front of the fraction.

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

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

Step 7

Simplify.

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Step 7.1

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 7.2

Multiply

\frac{2}{3}

$\frac{2}{3}$ by

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

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

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

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

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

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

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

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

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