Calculus Examples

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

Step 1

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

$\frac{d}{d x} [2 x^{3} x^{\frac{1}{3}}]$

Step 2

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

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Step 2.1

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

$\frac{d}{d x} [2 (x^{\frac{1}{3}} x^{3})]$

Step 2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [2 x^{\frac{1}{3} + 3}]$

Step 2.3

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

$\frac{d}{d x} [2 x^{\frac{1}{3} + 3 \cdot \frac{3}{3}}]$

Step 2.4

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

$\frac{d}{d x} [2 x^{\frac{1}{3} + \frac{3 \cdot 3}{3}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{d}{d x} [2 x^{\frac{1 + 3 \cdot 3}{3}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $3$ by $3$ .

$\frac{d}{d x} [2 x^{\frac{1 + 9}{3}}]$

Step 2.6.2

Add $1$ and $9$ .

$\frac{d}{d x} [2 x^{\frac{10}{3}}]$

Step 3

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{\frac{10}{3}}$ with respect to $x$ is $2 \frac{d}{d x} [x^{\frac{10}{3}}]$ .

$2 \frac{d}{d x} [x^{\frac{10}{3}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{10}{3}$ .

$2 (\frac{10}{3} x^{\frac{10}{3} - 1})$

Step 5

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

$2 (\frac{10}{3} x^{\frac{10}{3} - 1 \cdot \frac{3}{3}})$

Step 6

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

$2 (\frac{10}{3} x^{\frac{10}{3} + \frac{- 1 \cdot 3}{3}})$

Step 7

Combine the numerators over the common denominator.

$2 (\frac{10}{3} x^{\frac{10 - 1 \cdot 3}{3}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $3$ .

$2 (\frac{10}{3} x^{\frac{10 - 3}{3}})$

Step 8.2

Subtract $3$ from $10$ .

$2 (\frac{10}{3} x^{\frac{7}{3}})$

Step 9

Combine $\frac{10}{3}$ and $x^{\frac{7}{3}}$ .

$2 \frac{10 x^{\frac{7}{3}}}{3}$

Step 10

Combine $2$ and $\frac{10 x^{\frac{7}{3}}}{3}$ .

$\frac{2 (10 x^{\frac{7}{3}})}{3}$

Step 11

Multiply $10$ by $2$ .

$\frac{20 x^{\frac{7}{3}}}{3}$