Show full answers and steps to this problem **Problem 3:**Given that $ w = xyz + y^2 z + x + 1 $, $ x = e^t $, $ y = s^2 $, $ z = t + 1 $.(a) Find $\frac{\partial w}{\partial s}$.(b) Find $\frac{\partial w}{\partial t}$.(c) Find $\left( \frac{\partial w}{\partial t} \right) \bigg|_{t=0,s=0}$.(d) Find $\frac{\partial^2 w}{\partial x \partial y} $ ( = $ w_{xy} $ ).

Given that .(a) To find .(b) To find .(c) To find .Note: Since the student has posted many sub-parts…

Answered: 3. Given that w = xyz + y² + x + 1, x =…

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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**Problem 3:**

Given that $ w = xyz + y^2 z + x + 1 $, $ x = e^t $, $ y = s^2 $, $ z = t + 1 $.

(a) Find $\frac{\partial w}{\partial s}$.

(b) Find $\frac{\partial w}{\partial t}$.

(c) Find $\left( \frac{\partial w}{\partial t} \right) \bigg|_{t=0,s=0}$.

(d) Find $\frac{\partial^2 w}{\partial x \partial y} $ ( = $ w_{xy} $ ).

Expert Solution

Step 1: Define the problem.

Given that $straight w equals xyz plus straight y squared straight z plus straight x plus 1 comma space straight x equals straight e to the power of st comma space straight y equals straight s squared comma space straight z equals straight t plus 1$ .

(a) To find $fraction numerator partial differential straight w over denominator partial differential straight s end fraction$ .

(b) To find $fraction numerator partial differential straight w over denominator partial differential straight t end fraction$ .

(c) To find $right enclose fraction numerator partial differential straight w over denominator partial differential straight t end fraction end enclose subscript straight t equals 0 comma space straight s equals 0 end subscript$ .

Note: Since the student has posted many sub-parts of the question and does not mention any
specific part, so we are answering first three sub-parts according to our guidelines.
Student can re-post the remaining sub-part.