Q 15. Let聽t聽=聽f聽(蟿)聽be聽a聽diffe... [FREE SOLUTION]

Chapter 11: Q 15. (page 871)

$Let t = f (蟿) be a differentiable real - valued function of 蟿, and let r (t) be a differentiable vector function with either two or three components such that f (蟿) is in the domain of r for every value of 蟿 on some interval I . Find \frac{d (r (f (蟿)))}{d 蟿} .$

Short Answer

Expert verified

$\frac{d (r (f (蟿)))}{d 蟿} = r' (f (蟿)) . f' (蟿)$

Step by step solution

Step 1. Given information is:

$r (t) = <x (t), y (t), z (t)> be a differentiable vector function with three components . t = f (蟿) be a differentiable real - valued function of 蟿, and f (蟿) is in the domain of r for every value of 蟿 on some interval I .$

Step 2. Calculating drfτdτ

$\frac{d (r (f (蟿)))}{d 蟿} = \frac{d (r (t))}{d 蟿} (since f (蟿) = t) \frac{d (r (f (蟿)))}{d 蟿} = \frac{d <x (t), y (t), z (t)>}{d 蟿} (substitute for r (t)) \frac{d (r (f (蟿)))}{d 蟿} = <\frac{d (x (t))}{d 蟿}, \frac{d (y (t))}{d 蟿}, \frac{d (z (t))}{d 蟿}> \frac{d (r (f (蟿)))}{d 蟿} = <\frac{d x}{d t} . \frac{d t}{d 蟿}, \frac{d y}{d t} . \frac{d t}{d 蟿}, \frac{d z}{d t} . \frac{d t}{d 蟿}> \frac{d (r (f (蟿)))}{d 蟿} = <\frac{d x}{d t}, \frac{d y}{d t}, \frac{d z}{d t}> \frac{d t}{d 蟿} \frac{d (r (f (蟿)))}{d 蟿} = \frac{d r}{d t} . \frac{d t}{d 蟿} \frac{d (r (f (蟿)))}{d 蟿} = r' (t) . f' (蟿) \frac{d (r (f (蟿)))}{d 蟿} = r' (f (蟿)) . f' (蟿)$

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91影视

$Let t = f (蟿) be a differentiable real - valued function of 蟿, and let r (t) be a differentiable vector function with either two or three components such that f (蟿) is in the domain of r for every value of 蟿 on some interval I . Find \frac{d (r (f (蟿)))}{d 蟿} .$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating drfτdτ

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91影视

Lett=f(蟿)beadifferentiablereal-valuedfunctionof蟿,andletr(t)beadifferentiablevectorfunctionwitheithertwoorthreecomponentssuchthatf(蟿)isinthedomainofrforeveryvalueof蟿onsomeintervalI.Finddrf蟿d蟿.

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Calculating drfτdτ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

$Let t = f (蟿) be a differentiable real - valued function of 蟿, and let r (t) be a differentiable vector function with either two or three components such that f (蟿) is in the domain of r for every value of 蟿 on some interval I . Find \frac{d (r (f (蟿)))}{d 蟿} .$