Calculus Examples

$f (x) = \frac{1}{8} x^{2} \cdot (4 \sqrt{x})$

Step 1

Combine $\frac{1}{8}$ and $x^{2}$ .

$\frac{d}{d x} [\frac{x^{2}}{8} \cdot (4 \sqrt{x})]$

Step 2

Combine $4$ and $\frac{x^{2}}{8}$ .

$\frac{d}{d x} [\frac{4 x^{2}}{8} \cdot \sqrt{x}]$

Step 3

Combine $\frac{4 x^{2}}{8}$ and $\sqrt{x}$ .

$\frac{d}{d x} [\frac{4 x^{2} \sqrt{x}}{8}]$

Step 4

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

$\frac{d}{d x} [\frac{4 x^{2} x^{\frac{1}{2}}}{8}]$

Step 5

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

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Step 5.1

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

$\frac{d}{d x} [\frac{4 (x^{\frac{1}{2}} x^{2})}{8}]$

Step 5.2

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

$\frac{d}{d x} [\frac{4 x^{\frac{1}{2} + 2}}{8}]$

Step 5.3

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

$\frac{d}{d x} [\frac{4 x^{\frac{1}{2} + 2 \cdot \frac{2}{2}}}{8}]$

Step 5.4

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

$\frac{d}{d x} [\frac{4 x^{\frac{1}{2} + \frac{2 \cdot 2}{2}}}{8}]$

Step 5.5

Combine the numerators over the common denominator.

$\frac{d}{d x} [\frac{4 x^{\frac{1 + 2 \cdot 2}{2}}}{8}]$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $2$ by $2$ .

$\frac{d}{d x} [\frac{4 x^{\frac{1 + 4}{2}}}{8}]$

Step 5.6.2

Add $1$ and $4$ .

$\frac{d}{d x} [\frac{4 x^{\frac{5}{2}}}{8}]$

Step 6

Factor $4$ out of $4 x^{\frac{5}{2}}$ .

$\frac{d}{d x} [\frac{4 (x^{\frac{5}{2}})}{8}]$

Step 7

Cancel the common factors.

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Step 7.1

Factor $4$ out of $8$ .

$\frac{d}{d x} [\frac{4 x^{\frac{5}{2}}}{4 \cdot 2}]$

Step 7.2

Cancel the common factor.

$\frac{d}{d x} [\frac{4 x^{\frac{5}{2}}}{4 \cdot 2}]$

Step 7.3

Rewrite the expression.

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

Step 8

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 9

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 10

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 11

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

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

Step 12

Combine the numerators over the common denominator.

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

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 1$ by $2$ .

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

Step 13.2

Subtract $2$ from $5$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Multiply $2$ by $2$ .

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