Calculus Examples

$\sqrt{x \sqrt{x}}$

Step 1

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

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

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 $1$ .

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

Step 2.1.2

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Add $2$ and $1$ .

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{\frac{3}{2}}}$ as $x^{\frac{\frac{3}{2}}{2}}$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $2$ .

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

Step 6

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

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

Step 7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 8

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $4$ .

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

Step 10.2

Subtract $4$ from $3$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

Simplify.

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Step 12.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12.2

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

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