Calculus Examples

$f (x) = x \sqrt{x}$

Step 1

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

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Add $2$ and $1$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $3$ .

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

Step 8

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

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