Find the Derivative - d/dv square root of v

$\sqrt{v}$

Step 1

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

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

Step 2

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

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

Step 3

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

$\frac{1}{2} v^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 4

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

$\frac{1}{2} v^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} v^{\frac{1 - 1 \cdot 2}{2}}$

Step 6

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

Simplify.

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

$\frac{1}{2} \cdot \frac{1}{v^{\frac{1}{2}}}$

Multiply $\frac{1}{2}$ by $\frac{1}{v^{\frac{1}{2}}}$ .

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