Practice Problems on Derivative Computing (with Solutions)
This problem set is generated by Di. All of the problems came from past exams of Math 221. For
derivative computing – unlike many of other math concepts – more lectures do not help much, and
nothing compares to practicing on one’s own! The idea of this problem set is to get enough practice,
till the point that it becomes hard to make any mistake. :)
1. y =
2. y =
√
3
x2 sin x
tan x
x3 +2
3. y = x4 + sin x cos x
4. y =
5. y =
3
x3 −2x
x+3
√
x2 − 1 + 1
10
√
6. y = cos x2 tan x + 1
7. y = cos (cos (cos (3x)))
q
8. y = 1+x
2−x
√
9. y = x2 ( x + 2)
√
10. f (x) = 2 x2 + 1 + sin
11. h (x) =
4π
5
sin x
2x−3
12. y = x3 +
1
2x3
−
2
√
3x
13. y = sin (5x) cos (3x)
4
14. y = cos x2 + cos2 x
15. y = 12 + x cos x + x5
16. y = x2 + x − 1 sin x cos2 x
x
17. y = cos x2 + x+1
18. y =
x2 −2
x4 +1
√
19. y = x2 3 tan x
√
5
20. y =
x2 + 1 + x
21. y =
22. y =
√
1 − cos x (tan x)3
sin(x3 )
sin(x2 )
23. y = x3 + sin (x) cos2 (x)
1
24. y = x (10x + 6)2011
q
25. y = (sin x)3 + 1
26. y = tan x4 + 3x2 + 1
27. y =
sin(3x)
1+x4
3
28. y = (5 − 2 cos x) 2
√
80
29. y = (5 x + 3)
sin−4 (x) −
30. y =
1
x
31. y =
tan(2x)
(x+5)4
x
3
cos3 (x)
32. y = tan cos(x)
x
33. y = sin √xx2 +1
34. y = sin5 3x4 − 7x
The following problems involve ex and ln x, which we haven’t seen in our lecture so far, but we
will learn them later. We could practice them later when the material is covered. To do them, you
need to know:
1
1 d arctan x
=
.
(ex )0 = ex , (ln x)0 = ,
x
dx
1 + x2
1. f (x) = x ln e2x + 2
2. y = arctan e4x + 3x
√
3. y = ln x + x2 − 1
4. y = 3 ln (x sin x)
5. y = e− tan(x+1)
6. y = sin ln x + 3x2
2
7. y =
8. y =
ex
x+3
√
ln x + 1
√
9. y = e
x2 +1
10. y = ln
2(1+x2 )
x4
11. y = ln e2x
12. y = arctan (x − 1) +
13. y = x2 e
p
sin (ln x)
3x2 −5x
14. y = ln (4x + 6) e5x
2
Solutions:
3
1. y = x 2 sin x
3
1
y 0 = 32 x 2 sin x + x 2 cos x
2. y 0 =
sec2 x(x3 +2)−3x2 tan x
(x3 +2)2
2
3. y 0 = 3 x4 + sin x cos x
4x3 + cos2 x − sin2 x
4. y 0 =
(3x2 −2)(x+3)−(x3 −2x)
(x+3)2
10
1
x2 − 1 2 + 1
√
9
y 0 = 10
x2 − 1 + 1
5. y =
1
2
− 1
x2 − 1 2 2x
√
√
1
6. y 0 = − sin x2 2x tan x + 1 + cos x2 sec2
x + 1 12 (x + 1)− 2
7. y 0 = − sin (cos (cos (3x))) · (− sin (cos (3x))) · (− sin (3x)) · 3
8. y =
y0
=
1+x
2−x
1
2
1
2
1+x
2−x
− 1
2
(2−x)+(1+x)
(2−x)2
=
1
2
1+x
2−x
− 1
2
3
(2−x)2
1
9. y = x2 x 2 + 2
1
1
y 0 = 2x x 2 + 2 + x2 21 x− 2
1
10. f (x) = 2 x2 + 1 2 + sin
− 1
f 0 (x) = x2 + 1 2 2x
11. h0 (x) =
4π
5
tricky question! notice sin
4π
5
is just a constant.
cos x(2x−3)−2 sin x
(2x−3)2
1
12. y = x3 + 12 x−3 − 2x− 3
4
y 0 = 3x2 − 32 x−4 + 23 x− 3
13. y 0 = 5 cos (5x) cos (3x) + sin (5x) (− sin (3x)) 3
3
14. y 0 = 4 cos x2 + cos2 x
− sin x2 2x + 2 cos x (− sin x)
15. y 0 = cos x + x (− sin x) + 5x4
16. This problem is the product rule of 3 functions. Just kill each of them at one time. (abc)0 =
a0 bc + ab0 c + abc0 .
y 0 = (2x + 1) sin x cos2 x + x2 + x − 1 cos3 x + x2 + x − 1 sin x2 cos x (− sin x)
x
x+1−x
x
1
0
2
2
17. y = − sin x + x+1 2x + (x+1)2 = − sin x + x+1 2x + (x+1)2
18. y 0 =
2x(x4 +1)−4x3 (x2 −2)
(x4 +1)2
3
1
19. y = x2 (tan x) 3
2
1
y 0 = 2x (tan x) 3 + x2 31 (tan x)− 3 sec2 x
√
4 1
1
2 + 1 − 2 2x + 1
20. y 0 = 5
x2 + 1 + x
x
2
1
21. y = (1 − cos x) 2 (tan x)3
y0 =
22. y 0 =
1
2
1
1
(1 − cos x)− 2 sin x (tan x)3 + (1 − cos x) 2 3 (tan x)2 sec2 x
cos(x3 )3x2 sin(x2 )−cos(x2 )2x sin(x3 )
[sin(x2 )]2
23. y 0 = 3x2 + cos3 x + sin x · 2 cos x (− sin x)
24. y 0 = (10x + 6)2011 + x2011 (10x + 6)2010 10
1
2
25. y = (sin x)3 + 1
y0 =
1
2
(sin x)3 + 1
− 1
2
26. y 0 = sec2 x4 + 3x2 + 1
3 (sin x)2 cos x
4x3 + 6x
27. y 0 =
cos(3x)3(1+x4 )−4x3 sin(3x)
28. y 0 =
3
2
(1+x4 )2
1
(5 − 2 cos x) 2 (2 sin x)
√
79 5 − 1
2
29. y 0 = 80 (5 x + 3)
x
2
30. For the 1st term, you can also do quotient rule. Here I rewrite it as a product, so that I can
kill people. :)
y = x−1 sin−4 (x) −
x
3
cos3 x
y 0 = −x−2 sin−4 (x) + x−1 (−4) sin−5 (x) cos (x) − 31 cos3 x − x3 3 cos2 x (− sin x)
31. y 0 =
sec2 (2x)2(x+5)4 −4(x+5)3 tan(2x)
(x+5)8
32. y 0 = sec2
33. y 0 = cos
cos x
x
− sin x·x−cos x
√ x
x2 +1
x2
√x2 +1− 1
34. y 0 = 5 sin4 3x4 − 7x
2
(x2 +1)
−1
2
2x2
x2 +1
12x3 − 7
Well, I guess no one works till the last problem together with me... I admit computing derivatives
could be pretty boring and exhausted. But if you finished all of these problems, I believe derivative
computing will become part of your nature, and you can do them quickly and accurately. Now it’s
time to say: I came, I calculated, I conquered. :)
Some irrelevant aside by Di: Typing solutions is also very exhausted! Half way through typing,
I started to question myself why do I want to tortune myself on doing this extra amount of work.
4
lol Well... But if this is helpful to anyone in any sense, then it worths the time and effort. Also if
there’s any typo, please kindly let me know!
1. f 0 (x) = ln e2x + 2 +
2. y 0 =
1
1+(e4x +3x)2
x
e2x 2
e2x +2
4e4x + 3
−1
3. y 0 =
1+ 12 (x2 −1) 2 2x
√
x+ x2 −1
4. y 0 =
3
x sin x
(sin x + x cos x)
5. y 0 = e− tan(x+1) − sec2 (x + 1)
6. y 0 = cos ln x + 3x2 x1 + 6x
2
7. y 0 =
ex 2x(x+3)−ex
(x+3)2
8. y 0 =
1
2
2
1
(ln x + 1)− 2
√
1
x
− 1
x2 + 1 2 2x
10. y = ln 2 + ln 1 + x2 − 4 ln x
9. y 0 = e
x2 +1 1
2
Here we used the properties of logarithm that we’ll learn later:
a
ln (ab) = ln a + ln b, ln
= ln a − ln b.
b
y0 =
2x
1+x2
−
4
x
11. y = ln e2x = 2x
y0 = 2
12. y 0 =
1
1+(x−1)2
13. y 0 = 2xe3x
14. y 0 =
1
+ 21 (sin (ln x))− 2 cos (ln x) x1
2 −5x
4
5x
4x+6 e
+ x2 e3x
2 −5x
(6x − 5)
+ ln (4x + 6) e5x 5
5