Chapter 12: Q. 22 (page 964)

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. 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 $\frac{d z}{d t} = e^{\cos t} \cdot \sin^{2} t [3 \cos t + \sin^{2} t]$

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" $\frac{d z}{d t} = \frac{\partial z}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial z}{\partial y} \cdot \frac{d y}{d t}$

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

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

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. 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:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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