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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

01

Step 1. Given Information

The function is

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

02

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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Most popular questions from this chapter

Explain the difference between two antiderivatives of the function.

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

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Find the relationship between between the functions $g (x) + 10 and f (x)$ .

Find the critical points of f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f (x) = 2^{1 - I n x}$

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