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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use the Chain Rule to find ∂z/∂s and ∂z/∂t.

7. z = (xy)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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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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