Q. 22 Use Theorem 12.32 to find the in... [FREE SOLUTION]

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Chapter 12: Q. 22 (page 964)

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

Short Answer

Expert verified

The value is $\frac{d z}{d t} = e^{\cos t} 路 \sin^{2} t [3 \cos t + \sin^{2} t]$

Step by step solution

Step 1. Given Information:

Given:

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

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

Step 2. Solution:

We know that Theorem 12.32 state that

Given functions $z = f (x, y), x = u (t), a n d y = v (t)$ , for all values of t at which u and v are differentiable, and if f is differentiable at $(u (t), v (t))$ , then localid="1649687422575" $\frac{d z}{d t} = \frac{鈭� z}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� z}{鈭� y} 路 \frac{d y}{d t}$

localid="1649688445041" $U s i n g x = \sin t a n d y = \cos t i n z = x^{3} e^{y} w e g e t z = {(\sin t)}^{3} 路 e^{\cos t} z = e^{\cos t} 路 \sin^{3} t D i f f . w . r . t . t w e g e t \frac{d z}{d t} = e^{\cos t} \frac{d}{d t} \sin^{3} t + \sin^{3} t \frac{d}{d t} e^{\cos t} \frac{d z}{d t} = e^{\cos t} (3 \sin^{2} t 路 \cos t) + \sin^{3} t (e^{\cos t} 路 \sin t) \frac{d z}{d t} = e^{\cos t} 路 \sin^{2} t [3 \cos t + \sin^{2} t]$

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91影视

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

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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91影视

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.dzdtwhenz=x3ey,x=sint,andy=cost

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

Applied Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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