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Problem 1

Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \sqrt[3]{1 + 4x}$

Heather Z.

Problem 2

Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = (2x^3 + 5)^4$

Heather Z.

Problem 3

Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \tan \pi x$

Heather Z.

Problem 4

Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \sin(\cot x)$

Heather Z.

Problem 5

Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = e^{\sqrt{x}}$

Heather Z.

Problem 6

Write the composite function in the form $f(g(x)).$ [Identify the inner function $u = g(x)$ and the outer function $y = f(u).$ ] Then find the derivative $dy/ dx.$
$y = \sqrt{2 - e^x}$

Heather Z.

Problem 7

Find the derivative of the function.
$F(x) = (5x^6 + 2x^3)^4$

dd
Deepak D.

Problem 8

Find the derivative of the function.
$F(x) = (1 + x + x^2)^{99}$

PC
Partha Sarathi C.

Problem 9

Find the derivative of the function.
$f(x) = \sqrt{5x + 1}$

Heather Z.

Problem 10

Find the derivative of the function.
$f(x) = \frac {1}{\sqrt [3]{x^2 - 1}}$

Heather Z.

Problem 11

Find the derivative of the function.
$f(\theta) = \cos (\theta^2)$

Heather Z.

Problem 12

Find the derivative of the function.
$g(\theta) = \cos^2 \theta$

Heather Z.

Problem 13

Find the derivative of the function.
$y = x^2 e^{-3x}$

Heather Z.

Problem 14

Find the derivative of the function.
$f(t) = t \sin \pi t$

Heather Z.

Problem 15

Find the derivative of the function.
$f(t) = e^{at} \sin bt$

Heather Z.

Problem 16

Find the derivative of the function.
$g(x) = e^{x^2 - x}$

Heather Z.

Problem 17

Find the derivative of the function.
$f(x) = (2x - 3)^4 (x^2 + x + 1)^5$

Heather Z.

Problem 18

Find the derivative of the function.
$g(x) = (x^2 + 1)^3 (x^2 + 2)^6$

Heather Z.

Problem 19

Find the derivative of the function.
$h(t) = (t +1)^{2/3} (2t^2 - 1)^3$

Heather Z.

Problem 20

Find the derivative of the function.
$F(t) = (3t - 1)^4 (2t + 1)^{-3}$

Heather Z.

Problem 21

Find the derivative of the function.
$y = \sqrt \frac {x}{x + 1}$

Heather Z.

Problem 22

Find the derivative of the function.
$y = (x + \frac {1}{x})^5$

Heather Z.

Problem 23

Find the derivative of the function.
$y = e^{\tan \theta}$

Heather Z.

Problem 24

Find the derivative of the function.
$f(t) = 2^{t^3}$

Heather Z.

Problem 25

Find the derivative of the function.
$g(u) = ( \frac {u^3 - 1}{u^3 +1})^8$

Heather Z.

Problem 26

Find the derivative of the function.
$s(t) = \sqrt \frac {1 + \sin t}{1 + \cos t}$

Heather Z.

Problem 27

Find the derivative of the function.
$r(t) = 10^{2 \sqrt {t}}$

AK
Ayush K.

Problem 28

Find the derivative of the function.
$f(z) = e^{z/(z - 1)}$

Heather Z.

Problem 29

Find the derivative of the function.
$H(r) = \frac {(r^2 - 1)^3}{(2r + 1)^5}$

Heather Z.

Problem 30

Find the derivative of the function.
$J(\theta) = \tan^2 (n \theta)$

Heather Z.

Problem 31

Find the derivative of the function.
$F(t) = e^{t \sin 2t}$

Heather Z.

Problem 32

Find the derivative of the function.
$F(t) = \frac {t^2}{\sqrt {t^3 + 1}}$

Heather Z.

Problem 33

Find the derivative of the function.
$G(x) = 4^{C/x}$

Heather Z.

Problem 34

Find the derivative of the function.
$U(y) = (\frac {y^4 + 1}{y^2 + 1})^5$

Heather Z.

Problem 35

Find the derivative of the function.
$y = \cos (\frac {1 - e^{2x}}{1 + e^{2x}})$

Heather Z.

Problem 36

Find the derivative of the function.
$y = x^2 e^{-1/x}$

Heather Z.

Problem 37

Find the derivative of the function.
$y = \cot^2 (\sin \theta)$

Heather Z.

Problem 38

Find the derivative of the function.
$y = \sqrt {1 + xe^{-2x}}$

Heather Z.

Problem 39

Find the derivative of the function.
$f(t) = \tan (\sec(\cos t))$

Heather Z.

Problem 40

Find the derivative of the function.
$y = e^{\sin 2x} + \sin (e^{2x})$

Heather Z.

Problem 41

Find the derivative of the function.
$f(t) = \sin^2 (e^{\sin^2 t})$

Heather Z.

Problem 42

Find the derivative of the function.
$y = \sqrt {x + \sqrt {x + \sqrt {x}}}$

Heather Z.

Problem 43

Find the derivative of the function.
$g(x) = (2ra^{rx} + n)^P$

Heather Z.

Problem 44

Find the derivative of the function.
$y = 2^{3^{4^{x}}}$

Heather Z.

Problem 45

Find the derivative of the function.
$y = \cos \sqrt {\sin (\tan \pi x)}$

Heather Z.

Problem 46

Find the derivative of the function.
$y = [x + (x + \sin^2 x)^3]^4$

Heather Z.

Problem 47

Find $y^{\prime}$ and $y^{\prime \prime}$
$$y=\cos (\sin 3 \theta)$$

Heather Z.

Problem 48

Find $y'$ and $y".$
$y = \frac {1}{(1 + \tan x)^2}$

Frank L.

Problem 49

Find $y^{\prime}$ and $y^{\prime \prime}$
$$y=\sqrt{1-\sec t}$$

Frank L.

Problem 50

Find $y'$ and $y".$
$y = e^{e^x}$

Heather Z.

Problem 51

Find an equation of the tangent line to the curve at the given point.
$y = 2^x, (0, 1)$

Heather Z.

Problem 52

Find an equation of the tangent line to the curve at the given point.
$y = \sqrt {1 + x^3}, (2, 3)$

Heather Z.

Problem 53

Find an equation of the tangent line to the curve at the given point.
$y = \sin (\sin x), (\pi, 0)$

Heather Z.

Problem 54

Find an equation of the tangent line to the curve at the given point.
$y = xe^{-x^2}, (0, 0)$

Heather Z.

Problem 55

(a) Find an equation of the tangent line to the curve $y = 2/(1 + e^{-x})$ at the point (0, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Heather Z.

Problem 56

(a) The curve $y = \mid x \mid /\sqrt {2 - x^2}$ is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Heather Z.

Problem 57

(a) If $f(x) = x \sqrt {2 - x^2},$ find $f'(x).$
(b) Check to see that your answer to part (a) is reasonable by comparing the graph of $f$ and $f'$.

Heather Z.

Problem 58

The function $f(x) = \sin (x + \sin 2x), 0 \le x \le \pi,$ arises in applications to frequency modulation (FM) synthesis.
(a) Use a graph of $f$ produced by a calculator lo make a rough sketch of the graph of $f'.$
(b) Calculate $f'(x)$ and use this expression, with a calculator, to graph $f'.$ Compare with your sketch in part (a).

Heather Z.

Problem 59

Find all points on the graph of the function $f(x) = 2 \sin x + \sin^2 x$ at which the tangent line is horizontal.

Heather Z.

Problem 60

At what point on the curve $y = \sqrt {1 + 2x}$ is the tangent line perpendicular to the line $6x + 2y = 1?$

Heather Z.

Problem 61

If $F(x) = f(g(x)),$ where $f(-2) = 8, f'(-2) =4, f'(5) = 3, g(5) = -2,$ and $g'(5) = 6,$ find $F'(5).$

Heather Z.

Problem 62

If $h(x) = \sqrt {4 + 3f(x)},$ where $f(1) = 7$ and $f'(1) = 4,$ find $h'(1).$

Heather Z.

Problem 63

A table of values for $f, g, f' ,$ and $g'$ is given.
(a) If $h(x) = f(g(x)),$ find $h'(1).$
(b) If $H(x) = g(g(x)),$ find $H(1).$

Heather Z.

Problem 64

Let $f$ and $g$ be the function in Exercise 63.
(a) If $F(x) = f(f(x)),$ find $F'(2).$
(b) If $G(x) = g(g(x)),$ find $G'(3).$

Heather Z.

Problem 65

If $f$ and $g$ are the functions whose graphs are shown, let $u(x) = f(g(x)), v(x) = g(f(x)),$ and $w(x) = g(g(x)).$ Find each derivative, if it exists. If it does not exist, explain why.
(a) $u'(1)$
(b) $v'(1)$
(c) $w'(1)$

Carson M.

Problem 66

If $f$ is the function whose graph is shown, let $h(x) = f(f(x))$ and $g(x) = f(x^2).$ Use the graph of $f$ to estimate the value of each derivative,
(a) $h'(2)$
(b) $g'(2)$

Heather Z.