   Chapter 14.5, Problem 58E

Chapter
Section
Textbook Problem

Suppose that the equation F(x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two: z = f(x, y), y = g(x, z), x = h(y, z). If F is differentiable and Fx, Fy, and Fz are all nonzero, show that ∂ z ∂ x ∂ x ∂ y ∂ y ∂ z = − 1

To determine

To show: The equation zxxyyz=1 where z=f(x,y),y=g(x,z)andx=h(y,z) .

Explanation

Given:

The function F(x,y,z)=0 .

Where z=f(x,y),y=g(x,z)andx=h(y,z) .

Equation 7:zx=FxFz=FxFzandzy=FyFz=FyFz , where F is the function of x,yandz ”.

Calculation:

The value of zx using the equation 7 is as follows,

zx=FxFz

Consider the function F(x,y,z)=0 .

Substitute x=h(y,z) in the above equation and obtain the value of xy by using Chain Rule,

F(h(y,z),y,z)=0

Take partial derivative with respect to y in the above equation,

Fxxy+Fyyy+Fzzy=0Fxxy+Fy(1)+Fz(0)=0xy=FyFx

Thus, the value of xy=FyFx

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