   Chapter 14.5, Problem 7E

Chapter
Section
Textbook Problem

Use the Chain Rule to find ∂z/∂s and ∂z/∂t.7. z = (x − y)5, x = s2t, y = st2

To determine

To find: The values of zsandzt using Chain Rule if z=(xy)5 , x=s2t and y=st2 .

Explanation

Chain Rule:

“Suppose that z=f(x,y) is a differentiable function of x and y , where x=g(s,t) andy=h(s,t) are both differentiable functions of sandt . Then zs=zxxs+zyys and zt=zxxt+zyyt ”.

Calculation:

The functions z=(xy)5 . (1)

Take the partial derivative with respect to x of the equation (1),

zx=x(xy)5=5(xy)4(1)=5(xy)4

Thus, the partial derivate, zx=5(xy)4 .

Take the partial derivative with respect to y of the equation (1),

zy=5(xy)4(1)=5(xy)4

Thus, the partial derivate, zy=5(xy)4 .

Obtain the derivative of x with respect to s .

x=s2txs=2st=2st

Thus, the derivate, xs=2st .

Obtain the derivative of y with respect to s .

y=st2ys=(1)t2=t2

Thus, the derivate, ys=t2

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