The Chain Rule
As you work through the problems listed below, you should reference Chapter 2.2 of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource)
and your lecture notes.
EXPECTED SKILLS:
• Know how to use the chain rule to calculate derivatives of compositions of functions.
PRACTICE PROBLEMS:
For problems 1-16, calculate the derivative of the given function.
1. f (x) = (x3 + 4)−3
−9x2 (x3 + 4)−4
2. f (x) = (x2 + 2x)6
6(2x + 2)(x2 + 2x)5
3. f (x) =
√
x3 − 2
3x2
√
2 x3 − 2
1
4. f (x) = tan
x2
1
−3
2
−2x sec
x2
5. f (x) = sec 2x
2 sec (2x) tan (2x)
6. f (x) = cos3 3x
−9 sin (3x) cos2 (3x)
4
1
7. f (x) = x − 2
x
3 1
2
5
4
4 x − 2
5x + 3
x
x
5
1
8. f (x) =
x2 − 3
(3x − 5)3
2x
9(x2 − 3)
−
;
(3x − 5)3 (3x − 5)4
Video Solution: http://www.youtube.com/watch?v=jOBz GYnNWY
9. f (x) = (x2 + 2x)5 (x2 − 4x)3
5(2x + 2)(x2 + 2x)4 (x2 − 4x)3 + 3(2x − 4)(x2 − 4x)2 (x2 + 2x)5
π 10. f (x) = sin
x
π −πx−2 cos
x
11. f (x) = sin (sin 2x)
2 cos (sin 2x) cos 2x
12. f (x) = tan2 (x2 − 1)
4x tan (x2 − 1) sec2 (x2 − 1)
13. f (x) =
(x5
2
+ 4x3 − 4x)3
−6(5x4 + 12x2 − 4)
(x5 + 4x3 − 4x)4
14. f (x) =
x2 − 1
x2 + 1
3
12x(x2 − 1)2
(x2 + 1)4
15. y = 4x2 csc 5x
8x csc (5x) − 20x2 csc (5x) cot (5x)
16. y = tan (4 + x2 sin 3x)
3x2 cos 3x + 2x sin 3x sec2 4 + x2 sin 3x
2
17. Use the given table to calculate each of the following quantities:
x f (x) f 0 (x)
1 −2
−5
5
−3
2
3 −1
6
4
3
1
5
4
7
g(x) g 0 (x)
3
9
4
−2
7
−6
−2
5
1
8
d
[f (g(x))]
(a)
dx
x=2
−2
(b) (f ◦ g)0 (1)
54
d
(c)
[f (3x)]
dx
x=1
18
π i
d h √
g
x
(d)
2 sin
dx
4
x=3
−
9π
4
(e) h0 (2) if h(x) = x2 f (g(x))
4; Video Solution: http://www.youtube.com/watch?v=iJ9XL6xmnHI
For problems 18-20, calculate
d2 y
.
dx2
18. y = sin 3x
−9 sin 3x
2
1
19. y = x 1 +
x
2x−3
20. y =
1
1 − 2x
8
(1 − 2x)3
3
21. Suppose that f (x) is a twice differentiable function and define g(x) = x3 f (2x). Compute g 00 (x) in terms of f , f 0 , and f 00
g 00 (x) = 4x3 f 00 (2x) + 12x2 f 0 (2x) + 6xf (2x)
22. Let f (x) =
(x2
5
. Compute an equation of the tangent line to the graph of f (x)
+ 1)3
at x = 0.
y=5
23. Where does the tangent line to y = (5x + 7)3 at the point (−1, 8) cross the x-axis?
x=−
17
15
24. Find all points on the graph of y = sin2 x where the tangent lines are parallel to the
line y = x.
π
+ πk where k is any integer
4
25. What is the 100th derivative of y = sin (2x)?
2100 sin 2x
1
26. Multiple Choice: The derivative of y = x cos
is
x
1
1
2
− x sin
(a) 2x cos
x
x
2
1
(b) sin
x
x
1
(c) −2x sin
x
1
1
(d) 2x cos
+ sin
x
x
1
(e) sin
x
2
D
4