Calculus Examples

$\frac{2 u^{\frac{3}{2}}}{3}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $3$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Multiply.

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Step 9.1

Multiply $3$ by $2$ .

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

Step 9.2

Multiply $3$ by $2$ .

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

Step 10

Cancel the common factor.

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

Step 11

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

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