Stewart 9th Section 2.5: The Chain Rule

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Stewart 9th Section 2.5: The Chain Rule 0/88
1 Differentiation: Composite, Implicit, and Inverse Functions · Level 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 \(\dfrac{d y}{d x}\). \(y = (5 - x^4)^5\)
2 Differentiation: Composite, Implicit, and Inverse Functions · Level 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 \(\dfrac{d y}{d x}\). \(y = \sqrt{x^3 + 2}\)
3 Differentiation: Composite, Implicit, and Inverse Functions · Level 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 \(\dfrac{d y}{d x}\). \(y = \sin(\cos x)\)
4 Differentiation: Composite, Implicit, and Inverse Functions · Level 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 \(\dfrac{d y}{d x}\). \(y = \tan(x^2)\)
5 Differentiation: Composite, Implicit, and Inverse Functions · Level 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 \(\dfrac{d y}{d x}\). \(y = \sqrt{\sin x}\)
6 Differentiation: Composite, Implicit, and Inverse Functions · Level 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 \(\dfrac{d y}{d x}\). \(y = \sin(\sqrt{x})\)
7 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( f(x) = (2x^3 - 5x^2 + 4)^5 \)
8 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( f(x) = (x^5 + 3x^2 - x)^{50} \)
9 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( f(x) = \sqrt{5x + 1} \)
10 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( f(x) = \dfrac{1}{\sqrt{x^2 - 1}} \)
11 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( g(t) = \dfrac{1}{(2t + 1)^2} \)
12 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( F(t) = \left(\dfrac{1}{2t + 1}\right)^4 \)
13 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( A(t) = \dfrac{1}{(\cos t + \tan t)^2} \)
14 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( g(x) = (2 - \sin x)^{frac(3,2)} \)
15 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( f(\theta) = \cos(\theta^2) \)
16 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( g(\theta) = \cos^2 \theta \)
17 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( h(v) = v \sqrt[3]{1 + v^2} \)
18 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
\( f(t) = t \sin(\pi t) \)
19 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( F(x) = (4x + 5)^3 (x^2 - 2x + 5)^4 \)
20 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( G(z) = (1 - 4z)^2 \sqrt{z^2 + 1} \)
21 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( h(t) = (t + 1)^{frac(2,3)} (2t^2 - 1)^3 \)
22 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( F(t) = (3t - 1)^4 (2t + 1)^{-3} \)
23 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( y = \sqrt{\dfrac{x}{x + 1}} \)
24 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( y = \left(x + \dfrac{1}{x}\right)^5 \)
25 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( g(u) = \left(\dfrac{u^3 - 1}{u^3 + 1}\right)^8 \)
26 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( s(t) = \sqrt{\dfrac{1 + \sin t}{1 + \cos t}} \)
27 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( H(r) = \dfrac{(r^2 - 1)^3}{(2r + 1)^5} \)
28 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( F(t) = \dfrac{t^2}{\sqrt{t^3 + 1}} \)
29 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( y = \cos(\sec 4x) \)
30 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( J(\theta) = \tan^2(n \theta) \)
31 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( y = \dfrac{\cos x}{\sqrt{1 + \sin x}} \)
32 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( h(\theta) = \tan(\theta^2 \sin \theta) \)
33 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( y = \left(\dfrac{1 - \cos 2x}{1 + \cos 2x}\right)^4 \)
34 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( y = x \sin\left(\dfrac{1}{x}\right) \)
35 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( f(x) = \sin x \cos(1 - x^2) \)
36 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( y = \sin(t + \cos(\sqrt{t})) \)
37 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( F(t) = \tan(\sqrt{1 + t^2}) \)
38 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
\( G(z) = (1 + \cos^2 z)^3 \)
39 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( y = \sin^2(x^2 + 1) \)
40 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( g(u) = [(u^2 - 1)^6 - 3u]^4 \)
41 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( y = \cos^4(\sin^3 x) \)
42 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( y = \sin^3(\cos(x^2)) \)
43 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
\( f(t) = \tan(\sec(\cos t)) \)
44 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
\( y = \sqrt{x + \sqrt{x + \sqrt{x}}} \)
45 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
\( g(x) = (2r \sin(r x) + n)^p \)
46 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
\( y = \sin(\theta + \tan(\theta + \cos \theta)) \)
47 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
\( y = \cos(\sqrt{\sin(\tan(\pi x))}) \)
48 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
\( y = [x + (x + \sin^2 x)^3]^4 \)
49 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
Find \(y'\) and \(y''\). \(y = \cos(\sin 3 \theta)\)
50 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find \(y'\) and \(y''\). \(y = (1 + \sqrt{x})^3\)
51 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find \(y'\) and \(y''\). \(y = \sqrt{\cos x}\)
52 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find \(y'\) and \(y''\). \(y = \dfrac{4x}{\sqrt{x + 1}}\)
53 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find an equation of the tangent line to the curve at the given point. \(y = (3x - 1)^{-6}\), \((0, 1)\)
54 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find an equation of the tangent line to the curve at the given point. \(y = \sqrt{1 + x^3}\), \((2, 3)\)
55 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find an equation of the tangent line to the curve at the given point. \(y = \sin(\sin x)\), \((\pi, 0)\)
56 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find an equation of the tangent line to the curve at the given point. \(y = \sin^2 x \cos x\), \(\left(\dfrac{\pi}{2}, 0\right)\)
57 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
(a) Find an equation of the tangent line to the curve \(y = \tan\left(\dfrac{\pi x^2}{4}\right)\) at the point \((1, 1)\).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Enter your answer directly below each part above.

58 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
(a) The curve \(y = \dfrac{|x|}{\sqrt{2 - x^2}}\) 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.

Enter your answer directly below each part above.

59 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
(a) If \(f(x) = x \sqrt{2 - x^2}\), find \(f'(x)\).
(b) Check your answer to part (a) by comparing the graphs of \(f\) and \(f'\).

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60 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
The function \(f(x) = \sin(x + \sin 2x)\), \(0 \leq x \leq \pi\), arises in applications to frequency modulation (FM) synthesis.
(a) Use a graph of \(f\) produced by a calculator or computer to make a rough sketch of the graph of \(f'\).
(b) Calculate \(f'(x)\) and use this to graph \(f'\). Compare with your sketch in part (a).

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61 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Find all points on the graph of the function \(f(x) = 2 \sin x + \sin^2 x\) at which the tangent line is horizontal.
62 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
At what point on the curve \(y = \sqrt{1 + 2x}\) is the tangent line perpendicular to the line \(6x + 2y = 1\)?
63 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
If \(F(x) = f(g(x))\), where \(f(-2) = 8\), \(f'(-2) = 4\), \(f'(5) = 3\), \(g(5) = -2\), and \(g'(5) = 6\), find \(F'(5)\).
64 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
If \(h(x) = \sqrt{4 + 3f(x)}\), where \(f(1) = 7\) and \(f'(1) = 4\), find \(h'(1)\).
65 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
A table of values for \(f\), \(g\), \(f'\), and \(g'\) is given.
\(x\) \(f(x)\) \(g(x)\) \(f'(x)\) \(g'(x)\)
1 3 2 4 6
2 1 8 5 7
3 7 2 7 9
(a) If \(h(x) = f(g(x))\), find \(h'(1)\).
(b) If \(H(x) = g(f(x))\), find \(H'(1)\).

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66 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Let \(f\) and \(g\) be the functions in Exercise 65.
(a) If \(F(x) = f(f(x))\), find \(F'(2)\).
(b) If \(G(x) = g(g(x))\), find \(G'(3)\).

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67 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
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.
question image
(a) \(u'(1)\)
(b) \(v'(1)\)
(c) \(w'(1)\)

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68 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
If \(f\) is the function whose graph is shown, let \(h(x) = f(f(x))\) and \(g(x) = f(x^2)\). Use the graph to estimate each derivative.
question image
(a) \(h'(2)\)
(b) \(g'(2)\)

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69 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
If \(g(x) = \sqrt{f(x)}\), where the graph of \(f\) is shown, evaluate \(g'(3)\).
question image
70 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
Suppose \(f\) is differentiable on \(RR\) and \(\alpha\) is a real number. Let \(F(x) = f(x^\alpha)\) and \(G(x) = [f(x)]^\alpha\). Find expressions for (a) \(F'(x)\) and (b) \(G'(x)\).
71 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Let \(r(x) = f(g(h(x)))\), where \(h(1) = 2\), \(g(2) = 3\), \(h'(1) = 4\), \(g'(2) = 5\), and \(f'(3) = 6\). Find \(r'(1)\).
72 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
If \(g\) is a twice differentiable function and \(f(x) = x g(x^2)\), find \(f''\) in terms of \(g\), \(g'\), and \(g''\).
73 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
If \(F(x) = f(3f(4f(x)))\), where \(f(0) = 0\) and \(f'(0) = 2\), find \(F'(0)\).
74 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
If \(F(x) = f(x f(x f(x)))\), where \(f(1) = 2\), \(f(2) = 3\), \(f'(1) = 4\), \(f'(2) = 5\), and \(f'(3) = 6\), find \(F'(1)\).
75 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
Find the given derivative by finding the first few derivatives and observing the pattern that occurs. \(D^{103} \cos 2x\)
76 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
Find the given derivative by finding the first few derivatives and observing the pattern that occurs. \(D^{35} x \sin(\pi x)\)
77 Differentiation: Composite, Implicit, and Inverse Functions · Level 2
A vibrating string has displacement \(s(t) = 10 + \dfrac{1}{4} \sin(10 \pi t)\). Find the velocity of the string after \(t\) seconds.
78 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
An object moves along a horizontal line so that its position is given by \(s = A \cos(\omega t + \delta)\) (simple harmonic motion). Find the velocity of the object. When is the velocity 0?
79 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
The brightness of a Cepheid variable star is modeled by the function \(B(t) = 4.0 + 0.35 \sin\left(\dfrac{2 \pi t}{5.4}\right)\) where \(t\) is measured in days. Find the rate of change of brightness after \(t\) days.
80 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
In a model for the length of daylight (in hours) in Philadelphia on the \(t\)th day of the year: \(L(t) = 12 + 2.8 \sin[\dfrac{2 \pi}{365}(t - 80)]\) Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 \((t = 80)\) and May 21 \((t = 141)\).
81 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
A particle moves along a straight line with displacement \(s(t)\), velocity \(v(t)\), and acceleration \(a(t)\). Show that \(a(t) = v(t) \dfrac{d v}{d s}\). Explain the difference between the meanings of \(\dfrac{d v}{d t}\) and \(\dfrac{d v}{d s}\).
82 Differentiation: Composite, Implicit, and Inverse Functions · Level 3
Air is being pumped into a spherical weather balloon. At any time \(t\), the volume of the balloon is \(V(t)\) and its radius is \(r(t)\).
(a) What do the derivatives \(\dfrac{d V}{d r}\) and \(\dfrac{d V}{d t}\) represent?
(b) Express \(\dfrac{d V}{d t}\) in terms of \(\dfrac{d r}{d t}\).

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83 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
Use the Chain Rule to prove that the derivative of an even function is an odd function, and the derivative of an odd function is an even function.
84 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule. [Hint: Write \(\dfrac{f(x)}{g(x)} = f(x)[g(x)]^{-1}\).]
85 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
Use the Chain Rule to show that if \(\theta\) is measured in degrees, then \(\dfrac{d}{d \theta}(\sin \theta) = \dfrac{\pi}{180} \cos \theta\) (This gives one reason for the convention of using radian measure.)
86 Differentiation: Composite, Implicit, and Inverse Functions · Level 4
(a) Write \(|x| = \sqrt{x^2}\) and use the Chain Rule to show that \(\dfrac{d}{d x}|x| = \dfrac{x}{|x|}\).
(b) If \(f(x) = |\sin x|\), find \(f'(x)\) and sketch the graphs of \(f\) and \(f'\). Where is \(f\) not differentiable?
(c) If \(g(x) = \sin|x|\), find \(g'(x)\) and sketch the graphs of \(g\) and \(g'\). Where is \(g\) not differentiable?

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87 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
If \(F = f \circ g \circ h\) and \(f\), \(g\), and \(h\) are differentiable, show that \(F'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)\)
88 Differentiation: Composite, Implicit, and Inverse Functions · Level 5
If \(F = f \circ g\) and \(f\) and \(g\) are twice differentiable, show that \(F''(x) = f''(g(x)) \cdot [g'(x)]^2 + f'(g(x)) \cdot g''(x)\)

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