Calculus Examples

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

Step 1

Rewrite $\frac{d}{d x} [\frac{2 \sqrt{x^{3}}}{3}]$ as $\frac{d}{d x} [\frac{2 (x \sqrt{x})}{3}]$ .

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Step 1.1

Factor out $x^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.1

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

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Step 3.1.1

Raise $x$ to the power of $1$ .

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

Step 3.1.2

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

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

Step 3.2

Write $1$ as a fraction with a common denominator.

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

Step 3.3

Combine the numerators over the common denominator.

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

Step 3.4

Add $2$ and $1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $3$ .

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

Step 10

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

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

Step 11

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

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

Step 12

Multiply.

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Step 12.1

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{3 \cdot 2}$

Step 12.2

Multiply $3$ by $2$ .

$\frac{6 x^{\frac{1}{2}}}{6}$

Step 13

Cancel the common factor.

$\frac{6 x^{\frac{1}{2}}}{6}$

Step 14

Divide $x^{\frac{1}{2}}$ by $1$ .

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