Q. 36 Use Theorem 12.34 to find the in... [FREE SOLUTION]

91影视

Chapter 12: Q. 36 (page 961)

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.
$\frac{d 胃}{d t} when 胃 = \tan^{- 1} (\frac{yz}{x}), x = e^{t}, y = e^{2 t}, z = e^{3 t}$

Short Answer

Expert verified

The value of $\frac{d 胃}{d t} = \frac{4 e^{4 t}}{1 + e^{8 t}}$

Step by step solution

Step 1. Given Information.

$胃 = \tan^{- 1} (\frac{yz}{x}) x = e^{t} y = e^{2 t} z = e^{3 t}$

Step 2. Calculation.

By Theorem 12.34, we have

$\frac{d 胃}{d t} = \frac{鈭� 胃}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� 胃}{鈭� y} 路 \frac{d y}{d t} + \frac{鈭� 胃}{鈭� z} 路 \frac{d z}{d t} - - - - - - - (1)$

So first we find

$\frac{鈭� 胃}{鈭� x}, \frac{d x}{d t}, \frac{鈭� 胃}{鈭� y}, \frac{d y}{d t}, \frac{鈭� 胃}{鈭� z}, \frac{d z}{d t}$

So we have

$\frac{鈭� 胃}{鈭� x} = \frac{1}{1 + {(\frac{y z}{x})}^{2}} 路 (- \frac{y z}{x^{2}}) = - \frac{y z}{x^{2} + y^{2} z^{2}} \frac{d x}{d t} = e^{t} \frac{鈭� 胃}{鈭� y} = \frac{1}{1 + {(\frac{y z}{x})}^{2}} 路 (1 路 \frac{z}{x}) = \frac{z x}{x^{2} + y^{2} z^{2}} \frac{d y}{d t} = 2 e^{2 t} \frac{鈭� 胃}{鈭� z} = \frac{y x}{x^{2} + y^{2} z^{2}} \frac{d z}{d t} = 3 e^{3 t}$

Step 3. Calculation

Use these above values in (1) we get,

$\frac{d 胃}{d t} = \frac{鈭� 胃}{鈭� x} 路 \frac{d x}{d t} + \frac{鈭� 胃}{鈭� y} 路 \frac{d y}{d t} + \frac{鈭� 胃}{鈭� z} 路 \frac{d z}{d t} \frac{d 胃}{d t} = \frac{1}{x^{2} + y^{2} z^{2}} \{- y z e^{t} + 2 z x e^{2 t} + 3 y x e^{3 t}\}$

So from here, putting the value of $x, y, z$ in term of $t$ we get

$\frac{d 胃}{d t} = \frac{1}{x^{2} + y^{2} z^{2}} \{- y z e^{t} + 2 z x e^{2 t} + 3 y x e^{3 t}\} \frac{d 胃}{d t} = \frac{1}{e^{2 t} + e^{10 t}} \{- e^{5 t} 路 e^{t} + 2 e^{4 t} 路 e^{2 t} + 3 e^{3 t} 路 e^{3 t}\} \frac{d 胃}{d t} = \frac{4 e^{4 t}}{1 + e^{8 t}}$

Step 4. Conclusion.

The value of $\frac{d 胃}{d t} = \frac{4 e^{4 t}}{1 + e^{8 t}}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.
$\frac{d 胃}{d t} when 胃 = \tan^{- 1} (\frac{yz}{x}), x = e^{t}, y = e^{2 t}, z = e^{3 t}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Calculation

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.d胃dtwhen胃=tan-1yzx,x=et,y=e2t,z=e3t

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculation.

Step 3. Calculation

Step 4. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Applied Mathematics

Discrete Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use Theorem 12.34 to find the indicated derivatives in Exercises 31鈥�36. Be sure to simplify your answers.
$\frac{d 胃}{d t} when 胃 = \tan^{- 1} (\frac{yz}{x}), x = e^{t}, y = e^{2 t}, z = e^{3 t}$