Find the Derivative - d/dy z = square root of 9xy

$z = \sqrt{9 x y}$

Step 1

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

Rewrite $9 x y$ as $3^{2} (x y)$ .

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 2

Differentiate using the Constant Multiple Rule.

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

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

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

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{1}{2}}$ and $g (y) = x y$ .

To apply the Chain Rule, set $u$ as $x y$ .

$3 (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d y} [x y])$

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

$3 (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d y} [x y])$

Replace all occurrences of $u$ with $x y$ .

$3 (\frac{1}{2} {(x y)}^{\frac{1}{2} - 1} \frac{d}{d y} [x y])$

Step 4

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

$3 (\frac{1}{2} {(x y)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d y} [x y])$

Step 5

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

$3 (\frac{1}{2} {(x y)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d y} [x y])$

Step 6

Combine the numerators over the common denominator.

$3 (\frac{1}{2} {(x y)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d y} [x y])$

Step 7

Simplify the numerator.

Multiply $- 1$ by $2$ .

$3 (\frac{1}{2} {(x y)}^{\frac{1 - 2}{2}} \frac{d}{d y} [x y])$

Subtract $2$ from $1$ .

$3 (\frac{1}{2} {(x y)}^{\frac{- 1}{2}} \frac{d}{d y} [x y])$

Step 8

Combine fractions.

Move the negative in front of the fraction.

$3 (\frac{1}{2} {(x y)}^{- \frac{1}{2}} \frac{d}{d y} [x y])$

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

$3 (\frac{{(x y)}^{- \frac{1}{2}}}{2} \frac{d}{d y} [x y])$

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

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

Combine $\frac{1}{2 {(x y)}^{\frac{1}{2}}}$ and $3$ .

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

Step 9

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

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

Step 10

Combine fractions.

Combine $x$ and $\frac{3}{2 {(x y)}^{\frac{1}{2}}}$ .

$\frac{x \cdot 3}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d y} [y]$

Move $3$ to the left of $x$ .

$\frac{3 \cdot x}{2 {(x y)}^{\frac{1}{2}}} \frac{d}{d y} [y]$

Step 11

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

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

Step 12

Multiply $\frac{3 x}{2 {(x y)}^{\frac{1}{2}}}$ by $1$ .

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

Step 13

Simplify.

Apply the product rule to $x y$ .

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

Combine terms.

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

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

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

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

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

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

Raise $x$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

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

Find the Derivative - d/dy z = square root of 9xy

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