Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

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

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

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

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

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

So we have

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

Use these values in (1) we get

role="math" localid="1649695474446" $\frac{\partial z}{\partial t} = 2 x y^{3} \cdot \sin s + 3 x^{2} y^{2} \cdot (- s \sin t) \frac{\partial z}{\partial t} = 2 x y^{3} \cdot \sin s - 3 x^{2} y^{2} s \cdot \sin t$

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:

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

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Calculus

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Short Answer

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

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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∂t when z=x2y3, x=t sin s, and y=s cos t

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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.
$\frac{\partial z}{\partial t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$