   Chapter 14, Problem 38RE

Chapter
Section
Textbook Problem

Use a tree diagram to write out the Chain Rule for the case where w = f(t, u, v), t = t(p, q, r, s), u = u(p, q, r, s), and v = v(p, q, r, s) are all differentiable functions.

To determine

To write: The Chain Rule for the function w=f(t,u,v) , where t=t(p,q,r,s),u=u(p,q,r,s)andv=v(p,q,r,s) using tree diagram.

Explanation

The given function w=f(t,u,v) is differentiable.

Notice that t=t(p,q,r,s),u=u(p,q,r,s)andv=v(p,q,r,s) are the functions of p,q,rands .

The tree diagram can be drawn from the given functions as shown below in Figure 1.

From Figure 1, obtain the Chain Rules for wp , wq , wr and ws as follows,

wp=wt.tp+wu.up+wv

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