Short Answer

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.
$\frac{d y}{d t} w h e n y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$

The value is $\frac{d y}{d t} = t^{2} (\sqrt{t} \cdot \cos \sqrt{t} + 2 \sin \sqrt{t})$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information:

Given:

$y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$

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

Step 2. Solution:

$U s i n g r = t^{3} a n d θ = \sqrt{t} i n y = r \sin θ w e g e t y = t^{3} \sin \sqrt{t} D i f f . w . r . t . t w e g e t \frac{d y}{d t} = t^{3} \frac{d}{d t} \sin \sqrt{t} + \sin \sqrt{t} \frac{d}{d t} t^{3} \frac{d y}{d t} = t^{3} \cdot \cos \sqrt{t} \cdot \frac{1}{2 \sqrt{t}} + \sin \sqrt{t} \cdot 2 t^{2} \frac{d y}{d t} = t^{2} (\sqrt{t} \cdot \cos \sqrt{t} + 2 \sin \sqrt{t})$

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Calculus

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Short Answer

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.
$\frac{d y}{d t} w h e n y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$

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Step 1. Given Information:

Step 2. Solution:

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Calculus

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Short Answer

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.dydt when y=r sin θ, r=t3, and θ=t

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Step 2. Solution:

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Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.
$\frac{d y}{d t} w h e n y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$