Chapter 3: Q 36. (page 299)

Given that $u = u (t), v = v (t),$ and $w = w (t)$ are functions of $t$ and that $k$ is a constant, calculate the derivative $\frac{d f}{d t}$ of each function $f (t)$ . Your answers may involve $u, v, w,$ $\frac{d u}{d t}, \frac{d v}{d t}, \frac{d w}{d t}$ , $k$ , and/or $t$ .
$f (t) = \frac{k}{u^{2} w}$

Short Answer

Expert verified

The derivative is

$\frac{d f}{d t} = - \frac{2 k}{u^{3} w} \frac{d u}{d t} - \frac{k}{u^{2} w^{2}} \frac{d w}{d t}$

Step by step solution

Step 1. Given Information

The function is

$f (t) = \frac{k}{u^{2} w}$

Step 2. Finding the derivative

The function is

$f (t) = \frac{k}{u^{2} w}$

Differentiate both sides with respect to $t$

$\frac{d f}{d t} = \frac{d}{d t} (\frac{k}{u^{2} w})$

Take constant out of derivative

$\frac{d f}{d t} = k \frac{d}{d t} (\frac{1}{u^{2} w}) \frac{d f}{d t} = k \frac{d}{d t} (u^{- 2} w^{- 1})$

Apply product rule of derivative

$\frac{d f}{d t} = k [w^{- 1} \frac{d}{d t} (u^{- 2}) + u^{- 2} \frac{d}{d t} (w^{- 1})]$

Apply power and chain rule of derivative

$\frac{d f}{d t} = k [w^{- 1} (- 2 u^{- 3} \frac{d u}{d t}) + u^{- 2} (- 1 w^{- 2} \frac{d w}{d t})]$

Simplify it

$\frac{d f}{d t} = k [w^{- 1} (- 2 u^{- 3} \frac{d u}{d t}) + u^{- 2} (- 1 w^{- 2} \frac{d w}{d t})] \frac{d f}{d t} = k [- \frac{2}{u^{3} w} \frac{d u}{d t} - \frac{1}{u^{2} w^{2}} \frac{d w}{d t})] \frac{d f}{d t} = - \frac{2 k}{u^{3} w} \frac{d u}{d t} - \frac{k}{u^{2} w^{2}} \frac{d w}{d t}$

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Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the derivative

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Given that u=u(t),v=v(t),and w=w(t)are functions of tand that kis a constant, calculate the derivative dfdtof each function f(t). Your answers may involve u,v,w,dudt,dvdt,dwdt, k, and/or t.f(t)=ku2w

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Statistics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.