   Chapter 14, Problem 37RE

Chapter
Section
Textbook Problem

Suppose z = f(x, y), where x = g(s, t), y = h(s, t), g(1, 2) = 3, gs(1, 2) = −1, gt(1,2) = 4, h(1, 2) = 6, hs(l, 2) = −5, ht(1, 2) = 10, fx(3,6) = 7, and fy(3,6) = 8. Find ∂z/∂s and ∂z/∂t when s = 1 and t = 2.

To determine

To find: The value of zsandzt at s=1andt=2 using Chain Rule where z=f(x,y) x=g(s,t),y=h(s,t) and if g(1,2)=3,gs(1,2)=1,gt(1,2)=4,h(1,2)=6, hs(1,2)=5,ht(1,2)=10,fx(3,6)=7andfy(3,6)=8 .

Explanation

Chain Rule:

“Suppose that z=f(x,y) is a differentiable function of x and y , where x=g(t)andy=h(t) are both differentiable functions of t then, z is differentiable function of t and dzdt=zxdxdt+zydydt ”.

Calculation:

The value of zs using chain rule is as follows,

zs=zxxs+zyys (1)

The equation (1) at s=1 and t=2 can be expressed as,

zs|(1,2)=fx(3,6)gs(1,2)+fy(3,6)hs(1,2) (2)

Substitute the respective values in the equation (2),

zs|(1,2)=fx(3,6)gs(1,2)+fy(3,6)hs(1,2)=(7)(1)+8(5)=740=47

Thus, the value of zs at s=1andt=2 is 47

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