Find the Derivative - d/dy square root of 9y

$\sqrt{9 y}$

Step 1

Rewrite $\frac{d}{d y} [\sqrt{9 y}]$ as $\frac{d}{d y} [3 \sqrt{y}]$ .

Rewrite $9$ as $3^{2}$ .

$\frac{d}{d y} [\sqrt{3^{2} y}]$

Pull terms out from under the radical.

$\frac{d}{d y} [3 \sqrt{y}]$

Step 2

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

$\frac{d}{d y} [3 y^{\frac{1}{2}}]$

Step 3

Since $3$ is constant with respect to $y$ , the derivative of $3 y^{\frac{1}{2}}$ with respect to $y$ is $3 \frac{d}{d y} [y^{\frac{1}{2}}]$ .

$3 \frac{d}{d y} [y^{\frac{1}{2}}]$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

$3 \frac{y^{- \frac{1}{2}}}{2}$

Step 11

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

$\frac{3 y^{- \frac{1}{2}}}{2}$

Step 12

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

$\frac{3}{2 y^{\frac{1}{2}}}$