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Chapter 3: Q 33. (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 $t$ .
$f (t) = w {(u + t)}^{2}$

Short Answer

Expert verified

The derivative is

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

Step by step solution

01

Step 1. Given Information

The function is

$f (t) = w {(u + t)}^{2}$

02

Step 2. Finding the derivative

The function is

$f (t) = w {(u + t)}^{2}$

Differentiate both sides with respect to $t$

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

Apply product rule of derivative

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

Apply chain rule of derivative

localid="1648656938233" $\frac{d f}{d t} = {(u + t)}^{2} \frac{d w}{d t} + w 2 (u + t) \frac{d}{d t} (u + t) \frac{d f}{d t} = {(u + t)}^{2} \frac{d w}{d t} + w 2 (u + t) (\frac{d u}{d t} + \frac{d (t)}{d t}) \frac{d f}{d t} = {(u + t)}^{2} \frac{d w}{d t} + w 2 (u + t) (\frac{d u}{d t} + 1) \frac{d f}{d t} = {(u + t)}^{2} \frac{d w}{d t} + 2 w (u + t) (\frac{d u}{d t} + 1)$

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Most popular questions from this chapter

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f (x) = \frac{1 + x + x^{2}}{x^{2} + x - 2}$

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = e^{3 x} - e^{2 x}$

Explain the difference between two antiderivatives of the function.

For the graph of f in the given figure, approximate all the values x 鈭� (0, 4) for which the derivative of f is zero or does not exist. Indicate whether f has a local maximum, minimum, or neither at each of these critical points.

Determine whether or not each function $f$ satisfies the hypotheses of the Mean Value Theorem on the given interval $[a, b]$ . For those that do, use derivatives and algebra to find the exact values of all $c 鈭� (a, b)$ that satisfy the conclusion of the Mean Value Theorem.

$f (x) = \tan (x), [a, b] = [- 蟺, 蟺]$ .

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