Chapter 12: Q. 27 (page 964)

Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.
$\frac{\partial z}{\partial s} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Short Answer

Expert verified

The value is $\frac{\partial z}{\partial s} = s^{2} t^{2} \cdot \sin s \cdot \cos^{3} t (2 s \cos t + 3 \sin s)$

Step by step solution

Step 1. Given Information:

Given:

$z = x^{2} y^{3}, x = t \sin s a n d y = s \cos t$

We have to find the indicated derivatives and express your answers as functions of a single variable.

Step 2. Solution:

By Theorem 12.33, we have

$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial s} - - - (1)$

So first we find $\frac{\partial z}{\partial x}, \frac{\partial x}{\partial s}, \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial s}$

So we have

$\frac{\partial z}{\partial x} = 2 x y^{3} \frac{\partial z}{\partial y} = 3 x^{2} y^{2} \frac{\partial x}{\partial s} = t \cos s \frac{\partial y}{\partial s} = \cos t$

Use these values in (1) we get

localid="1649693736225" $\frac{\partial z}{\partial s} = 2 x y^{3} \cdot t \cdot \cos t + 3 x^{2} y^{2} \cdot c o s t$

This result is correct, but it is preferable to write the function as a function of just s and t.

We use $x = t \sin s a n d y = s \cos t$ to do so:

localid="1649695745634" role="math" $\frac{\partial z}{\partial s} = 2 (t \sin s) {(s \cos t)}^{3} \cdot t \cdot \cos t + 3 {(t \sin s)}^{2} {(s \cos t)}^{2} \cdot c o s t \frac{\partial z}{\partial s} = 2 s^{3} t^{2} \cdot \sin s \cdot \cos^{4} t + 3 s^{2} t^{2} \sin^{2} s \cdot \cos^{3} t \frac{\partial z}{\partial s} = s^{2} t^{2} \cdot \sin s \cdot \cos^{3} t (2 s \cos t + 3 \sin s)$

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Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.
$\frac{\partial z}{\partial s} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.∂z∂swhenz=x2y3,x=tsins,andy=scost

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Discrete Mathematics

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Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use Theorem 12.33 to find the indicated derivatives in Exercises 27–30. Express your answers as functions of two variables.
$\frac{\partial z}{\partial s} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$