# Suppose z = x^2 sin y, x = 3 s^2 + 3 t^2, y = 2 s t. A. Use the Chain Rule to find partial z /...

## Question:

Suppose {eq}z=x^2\sin y,x=3s^2+3t^2,y=2st {/eq}.

A. Use the Chain Rule to find {eq}\frac{\partial z}{\partial s} {/eq} and {eq}\frac{\partial z}{\partial t} {/eq} as functions of {eq}x,y,s {/eq} and {eq}t {/eq}.

{eq}\frac{\partial z}{\partial s}= {/eq}

{eq}\frac{\partial z}{\partial t}= {/eq}

B. Find the numerical values of {eq}\frac{\partial z}{\partial s} {/eq} and {eq}\frac{\partial z}{\partial t} {/eq} when {eq}(s,t)=(3,1) {/eq}.

{eq}\frac{\partial z}{\partial s}(3,1)= {/eq}

{eq}\frac{\partial z}{\partial t}(3,1)= {/eq}

## Chain Rule:

Suppose {eq}z=f(u,v) {/eq} and {eq}u {/eq} and {eq}v {/eq} are both functions of {eq}x {/eq} and {eq}y. {/eq} Then we can view {eq}z {/eq} as a function of {eq}x {/eq} and {eq}y {/eq} and compute the partial derivatives using the chain rule below.

{eq}\dfrac{\partial z}{\partial x}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial x}+\dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial x} {/eq}

and

{eq}\dfrac{\partial z}{\partial y}=\dfrac{\partial z}{\partial u}\dfrac{\partial u}{\partial y}+\dfrac{\partial z}{\partial v}\dfrac{\partial v}{\partial y}. {/eq}