   Chapter 14.5, Problem 18E

Chapter
Section
Textbook Problem

Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable.18. w = f(x, y, z), where x = x(u, v), y = y(u, v), z = z(u, v)

To determine

To write: The Chain Rule for the function w=f(x,y,z) , where x=x(u,v),y=y(u,v)andz=z(u,v) using tree diagram.

Explanation

The given function w=f(x,y,z) is differentiable.

Notice that x=x(u,v),y=y(u,v)andz=z(u,v) are the functions of uandv .

The tree diagram can be drawn for the given function as shown below in Figure 1.

From Figure 1, obtain the Chain Rules for wu and wv as follows.

wu=wx

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