Calculus Examples

$\frac{\sqrt{x^{5}}}{2}$

Step 1

Rewrite $\frac{d}{d x} [\frac{\sqrt{x^{5}}}{2}]$ as $\frac{d}{d x} [\frac{x^{2} \sqrt{x}}{2}]$ .

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Step 1.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 1.1.1

Factor out $x^{4}$ .

$\frac{d}{d x} [\frac{\sqrt{x^{4} x}}{2}]$

Step 1.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $2$ .

$\frac{d}{d x} [\frac{x^{\frac{4 + 1}{2}}}{2}]$

Step 3.5.2

Add $4$ and $1$ .

$\frac{d}{d x} [\frac{x^{\frac{5}{2}}}{2}]$

Step 4

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{\frac{5}{2}}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{\frac{5}{2}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$\frac{1}{2} (\frac{5}{2} x^{\frac{5}{2} - 1})$

Step 6

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

$\frac{1}{2} (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 7

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

$\frac{1}{2} (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 8

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}})$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} (\frac{5}{2} x^{\frac{5 - 2}{2}})$

Step 9.2

Subtract $2$ from $5$ .

$\frac{1}{2} (\frac{5}{2} x^{\frac{3}{2}})$

Step 10

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$\frac{1}{2} \cdot \frac{5 x^{\frac{3}{2}}}{2}$

Step 11

Multiply $\frac{1}{2}$ by $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{5 x^{\frac{3}{2}}}{2 \cdot 2}$

Step 12

Multiply $2$ by $2$ .

$\frac{5 x^{\frac{3}{2}}}{4}$