# Use the Product Rule or the Quotient Rule to find the derivative of the given function. Z(y)=...

## Question:

Use the Product Rule or the Quotient Rule to find the derivative of the given function.

{eq}Z(y)= \frac{4y-y^2}{6-y} {/eq}

## Quotient Rule:

The derivative of a function determines the slope of the tangent line to the curve and also the rate of change of one variable w.r.t. another variable. The quotient rule of differentiation of the ratio of two functions {eq}f \& g {/eq} is as follows:

{eq}\boxed{\dfrac{dy}{dx}\dfrac{f}{g}=\dfrac{f'g-g'f}{g^2}} {/eq}

• {eq}f \ \& \ g {/eq} are the functions of {eq}x. {/eq}

Given: {eq}Z(y)= \dfrac{4y-y^2}{6-y} {/eq}

Differentiating both sides w.r.t. {eq}y {/eq}, using the quotient rule, we get:

• {eq}Z'(y)=\dfrac{(6-y)\dfrac{d}{dy}(4y-y^2)-(4y-y^2)\dfrac{d}{dy}(6-y)}{(6-y)^2}\\\Rightarrow Z'(y)=\dfrac{(6-y)(4-2y)+(4y-y^2)}{(6-y)^2}\\\Rightarrow Z'(y)=\dfrac{24-12y-4y+2y^2+4y-y^2}{(6-y)^2}\\\Rightarrow Z'(y)=\dfrac{y^2-12y+24}{(6-y)^2} {/eq}