Calculus Examples

$2 \sqrt{u}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Move $u^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 12

Cancel the common factor.

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

Step 13

Rewrite the expression.

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