Calculus Examples

$\sqrt[3]{x^{4}}$

Step 1

Rewrite

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

$\frac{d}{d x} [x \sqrt[3]{x}]$ .

Tap for more steps...

Step 1.1

Factor out

$x^{3}$ .

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

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [x \sqrt[3]{x}]$

Step 2

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt[3]{x}$ as

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

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

Step 3

Multiply

$x$ by

$x^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 3.1

Multiply

$x$ by

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

Tap for more steps...

Step 3.1.1

Raise

$x$ to the power of

$1$ .

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

Step 3.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2

Write

$1$ as a fraction with a common denominator.

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add

$3$ and

$1$ .

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

Step 4

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = \frac{4}{3}$ .

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

Step 5

To write

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

$\frac{3}{3}$ .

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

Step 6

Combine

$- 1$ and

$\frac{3}{3}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply

$- 1$ by

$3$ .

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

Step 8.2

Subtract

$3$ from

$4$ .

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

Step 9

Combine

$\frac{4}{3}$ and

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

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