   Chapter 14.5, Problem 10E

Chapter
Section
Textbook Problem

Use the Chain Rule to find ∂z/∂s and ∂z/∂t.10. z = x e x y ,   x = 1 + s t ,   y = s 2 − t 2

To determine

To find: The values of zsandzt using Chain Rule if z=xexy , x=1+st and y=s2t2 .

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 given function is z=xexy (1)

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

zx=x(y)(exy)+exy(12x)

Thus, the partial derivate, zx=yxexy+12xexy .

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

zy=x(x)(exy)=xexyx

Thus, the partial derivate, zy=xexyx .

Obtain the partial derivative of x with respect to s .

x=1+stxs=0+(1)t=t

Thus, the partial derivate, xs=t .

Obtain the partial derivative of y with respect to s .

y=s2t2ys=2s0=2s

Thus, the partial derivate, ys=2s

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