Hello,Can someone please show how to solve this problem?Thank you Problem 5.5 Chain ruleConsider the functionw = 4e* ln(yz),with intermediate variables x, y, and z, which depend on the independent variables u and v via:x = ln(u cos v),y = usin v,z=u² + v,дwa) Express as a function of u and v.dub) EvaluateθωJuat (u, v) = (π/4, π/4).

Answered: Problem 5.5 Chain rule Consider the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Hello,

Can someone please show how to solve this problem?

Thank you

Problem 5.5 Chain rule
Consider the function
w = 4e* ln(yz),
with intermediate variables x, y, and z, which depend on the independent variables u and v via:
x = ln(u cos v),
y = usin v,
z=u² + v,
дw
a) Express as a function of u and v.
du
b) Evaluate
θω
Ju
at (u, v) = (π/4, π/4).

Expert Solution

Step 1

It is given that $w = 4 e^{x} \ln (y z)$ , $x = \ln (u \cos v)$ , $y = u \sin v$ and $z = u^{2} + v$ .

(a) The objective is to use the chain rule to evaluate $\frac{\partial w}{\partial u}$ as a function of $u$ and $v$ .

(b) The objective is to evaluate $\frac{\partial w}{\partial u}$ at $(u, v) = (\frac{π}{4}, \frac{π}{4})$ .

The chain rule states that $\frac{\partial w}{\partial u} = \frac{\partial w}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial w}{\partial y} \cdot \frac{\partial y}{\partial u} + \frac{\partial w}{\partial z} \cdot \frac{\partial z}{\partial u} \dots (1)$ .