# Find the derivative of the function. y=\sqrt{xy+9}

## Question:

Find the derivative of the function.

{eq}y =\sqrt{xy + 9} {/eq}

## Differentiation:

In such question direst apply formula that particular function.

So, as question have square root function,

Formula to be applied is :

{eq}(\sqrt x) = \frac{1}{2 \sqrt x} {/eq}

## Answer and Explanation:

We have,

{eq}y =\sqrt{xy + 9} {/eq}

now,

Differentiating both sides,

{eq}y' = (\sqrt{xy + 9})' {/eq}

Using Formula,

{eq}\displaystyle y' = \frac{1}{2 \sqrt{xy + 9}} (xy + 9)' \\ \displaystyle y' = \frac{1}{2 \sqrt{xy + 9}} (y + xy') \\ \displaystyle y' - \frac{xy'}{2 \sqrt{xy + 9}}= \frac{y}{2 \sqrt{xy + 9}} \\ \displaystyle y' (\frac{2 \sqrt{xy + 9} - x}{2 \sqrt{xy + 9}}= \frac{y}{2 \sqrt{xy + 9}} \\ \displaystyle y' = \frac{y}{2 \sqrt{xy + 9} - x} \\ {/eq}

so,

{eq}\therefore \color{blue}{\displaystyle y' = \frac{y}{2 \sqrt{xy + 9} - x}} {/eq}

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