Calculus Examples

$\frac{x^{\frac{4}{3}}}{4}$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x^{\frac{4}{3}}}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{\frac{4}{3}}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{\frac{4}{3}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$\frac{1}{4} (\frac{4}{3} x^{\frac{4}{3} - 1})$

Step 3

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

$\frac{1}{4} (\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}})$

Step 4

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

$\frac{1}{4} (\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}})$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{4} (\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$\frac{1}{4} (\frac{4}{3} x^{\frac{4 - 3}{3}})$

Step 6.2

Subtract $3$ from $4$ .

$\frac{1}{4} (\frac{4}{3} x^{\frac{1}{3}})$

Step 7

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

$\frac{1}{4} \cdot \frac{4 x^{\frac{1}{3}}}{3}$

Step 8

Multiply $\frac{1}{4}$ by $\frac{4 x^{\frac{1}{3}}}{3}$ .

$\frac{4 x^{\frac{1}{3}}}{4 \cdot 3}$

Step 9

Multiply $4$ by $3$ .

$\frac{4 x^{\frac{1}{3}}}{12}$

Step 10

Factor $4$ out of $4 x^{\frac{1}{3}}$ .

$\frac{4 (x^{\frac{1}{3}})}{12}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $4$ out of $12$ .

$\frac{4 x^{\frac{1}{3}}}{4 \cdot 3}$

Step 11.2

Cancel the common factor.

$\frac{4 x^{\frac{1}{3}}}{4 \cdot 3}$

Step 11.3

Rewrite the expression.

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