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Derivative Practice
Mathematics 1 (AMATH1S)
Vaal University of Technology
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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. :)
- y=
√ 3
x 2 sinx
y=tanx x 3 +
y=
(
x 4 + sinxcosx
) 3
- y=x
3 − 2 x x+
- y=
(√
x 2 −1 + 1
) 10
- y= cos
(
x 2
)
tan
(√
x+ 1
)
y= cos (cos (cos (3x)))
y=
√
1+x 2 −x
- y=x 2 (
√
x+ 2)
10(x) = 2
√
x 2 + 1 + sin
( 4 π 5
)
11(x) = 2 sinx−x 3
12=x 3 + 2 x 13 −√ 32 x
13= sin (5x) cos (3x)
14=
(
cos
(
x 2
)
- cos 2 x
) 4
15= 12 +xcosx+x 5
16=
(
x 2 +x− 1
)
sinxcos 2 x
17= cos
(
x 2 +xx+
)
18=x
2 − 2 x 4 +
19=x 2
√ 3
tanx
20=
(√
x 2 + 1 +x
) 5
21=
√
1 −cosx(tanx)
3
22=
sin(x 3 ) sin(x 2 )
23=x 3 + sin (x) cos 2 (x)
24=x(10x+ 6)
2011
25=
√
(sinx)
3 + 1
26= tan
(
x 4 + 3x 2 + 1
)
27=
sin(3x) 1+x 4
28= (5−2 cosx)
3 2
29= (
√
x+ 3)
80
30=
1 xsin
− 4 (x)−
x 3 cos
3 (x)
31=
tan(2x) (x+5) 4
32= tan
(
cos(x) x
)
33= sin
(
√x x 2 +
)
34= sin 5
(
3 x 4 − 7 x
)
The following problems involveexand lnx, which we haven’t seen in our lecture so far, but we
will learn them later. We could practice them later when the materialis covered. To do them, you
need to know:
(e
x )
′ =e
x ,(lnx)
′ =
1
x
,
darctanx
dx
=
1
1 +x 2
.
- f(x) =xln
(
e 2 x+ 2
)
- y= arctan
(
e 4 x+ 3x
)
- y= ln
(
x+
√
x 2 − 1
)
y= 3 ln (xsinx)
y=e−tan(x+1)
y= sin
(
lnx+ 3x 2
)
- y= e
x 2 x+
- y=
√
lnx+ 1
- y=e
√ x 2 +
10= ln
(
2 (1+x 2 ) x 4
)
11= lne 2 x
12= arctan (x−1) +
√
sin (lnx)
13=x 2 e 3 x
2 − 5 x
14= ln (4x+ 6)e 5 x
19=x 2 (tanx)
1 3
y′= 2x(tanx)
1 3 +x 21 3 (tanx)
− 23 sec 2 x
20′= 5
(√
x 2 + 1 +x
) 4 (
1 2
(
x 2 + 1
)− 1
22 x+ 1
)
21= (1−cosx)
1 2 (tanx) 3
y′=
1 2 (1−cosx)
− 12 sinx(tanx)
3 + (1−cosx)
1 2 3 (tanx) 2 sec 2 x
22′=
cos(x 3 ) 3 x 2 sin(x 2 )−cos(x 2 ) 2 xsin(x 3 ) [sin(x 2 )] 2
23′= 3x 2 + cos 3 x+ sinx·2 cosx(−sinx)
24′= (10x+ 6)
2011 +x2011 (10x+ 6)
2010 10
25=
(
(sinx)
3 + 1
) 1
2
y′= 12
(
(sinx)
3 + 1
)− 1
2 3 (sinx)
2 cosx
26′= sec 2
(
x 4 + 3x 2 + 1
) (
4 x 3 + 6x
)
27′=
cos(3x)3(1+x 4 )− 4 x 3 sin(3x) (1+x 4 ) 2
28′= 32 (5−2 cosx)
1 2 (2 sinx)
29′= 80 (
√
x+ 3)
79
(
5 2 x
− 12
)
- 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 cos 3 x
y′=−x− 2 sin− 4 (x) +x− 1 (−4) sin− 5 (x) cos (x)− 13 cos 3 x−x 3 3 cos 2 x(−sinx)
31′=
sec 2 (2x)2(x+5) 4 −4(x+5) 3 tan(2x) (x+5) 8
32′= sec 2
(cosx x
)−sinx·x−cosx x 2
33′= cos
(
√x x 2 +
)√
x 2 +1− 12 (x 2 +1)
− 12 2 x 2 x 2 +
34′= 5 sin 4
(
3 x 4 − 7 x
) (
12 x 3 − 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 andaccurately. 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.
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!
- f′(x) = ln
(
e 2 x+ 2
)
+e 2 xx+2e 2 x 2
- y′= 1 1+(e 4 x+3x) 2
(
4 e 4 x+ 3
)
- y′=
1+ 12 (x 2 − 1 )
− 12 2 x x+
√ x 2 − 1
y′=xsin 3 x(sinx+xcosx)
y′=e−tan(x+1)
(
−sec 2 (x+ 1)
)
- y′= cos
(
lnx+ 3x 2
) ( 1
x+ 6x
)
- y′=
ex
2 2 x(x+3)−ex
2
(x+3) 2
- y′= 12 (lnx+ 1)
− 121 x
- y′=e
√ x 2 +1 1 2
(
x 2 + 1
)− 1
22 x
10= ln 2 + ln
(
1 +x 2
)
−4 lnx
Here we used the properties of logarithm that we’ll learn later:
ln (ab) = lna+ lnb,ln
(
a
b
)
= lna−lnb.
y′=
2 x 1+x 2 −
4 x
11= lne 2 x= 2x
y′= 2
12′= 1 1+(x−1) 2
- 12 (sin (lnx))
− 12 cos (lnx) 1 x
13′= 2xe 3 x
2 − 5 x +x 2 e 3 x
2 − 5 x (6x−5)
14′= 4 x 4 +6e 5 x+ ln (4x+ 6)e 5 x 5
Derivative Practice
Course: Mathematics 1 (AMATH1S)
University: Vaal University of Technology
