Calculus 2 These are questions to practice in calculus on differentiation

1. 100 Derivatives
Great for calc 1 and calc 2 students
Video: https://youtu.be/AegzQ_dip8k
©blackpenredpen
July 20th, 2019
!1

2. (Q1.) !
(A) ! (B) ! (C) !
(Q2.) !
(A) ! (B) ! (C) !
(Q3.) !
(A) ! (B) ! (C) !
(Q4.) !
(A) ! (B) ! (C) !
(Q5.) !
(A) ! (B) ! (C) !
(Q6.) !
(A) ! (B) ! (C) !
(Q7.) !
(A) ! (B) ! (C) !
(Q8.) !
(A) ! (B) ! (C) !
(Q9.) !
(A) ! (B) ! (C) !
d
dx
ax2
+bx +c
( )
2ax +b 2ax +b +c 2a +b +c
d
dx
sinx
1+cosx
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
cosx
(1+cosx)2
1
1+cosx
sinx
(1+cosx)2
d
dx
1+cosx
sinx
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎟
−cscx cot x +sec2
x cosx +cot x −cscx cot x −csc2
x
d
dx
3x +1
( )
3
2 3x +1
1
2 3x +1
3
3x +1
d
dx
sin3
(x)+sin x3
( )
( )
3sin2
x +cos(x3
) 3sin2
x cosx +3x2
cos(x3
) 3sin2
x +cos(3x2
)
d
dx
1
x4
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎟
−4
x8
1
4x3
−4
x5
d
dx
(1+cot x)3
( )
−3csc2
x (1+cot x)2
−3csc2
x (1−csc2
x)2
3(1−csc2
x)2
d
dx
x2
2x3
+1
( )
10
( )
2x(2x3
+1)9
(2x3
+5x +1) 2x(2x3
+1)9
(32x3
+1) 120x3
(2x3
+1)9
d
dx
x
(x2
+1)2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2x +3
(x2
+1)4
1
2(x +1)
−3x2
+1
(x2
+1)3
!2

3. (Q10.) !
(A) ! (B) ! (C) !
(Q11.) !
(A) ! (B) ! (C) !
(Q12.) !
(A) ! (B) ! (C) !
(Q13.) !
(A) ! (B) ! (C) !
(Q14.) !
(A) ! (B) ! (C) !
(Q15.) !
(A) ! (B) ! (C) !
(Q16.) !
(A) ! (B) ! (C) !
(Q17.) !
(A) ! (B) ! (C) !
d
dx
20
1+5e−2x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
200e−2x
1+5e−2x
( )
2
200
1+5e−2x
20e−2x
1+5e−2x
( )
2
d
dx
ex
+e x
( )
ex
+ xe x ex
2
+
e x
2 x
1
2 ex
+
e x
2 x
d
dx
sec3
(2x)
( )
sec3
(2x)tan2
(2x) 3sec2
(2x)tan(2x) 6sec3
(2x)tan(2x)
d
dx
1
2 secx tanx + 1
2 ln secx + tanx
( )
( )
sec3
x 2sec2
x tanx 3sec3
x tanx
d
dx
xex
1+ ex
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
xex
+ex
−e2x
(1+ex
)2
xex
+ex
+e2x
(1+ex
)2
xex
+ex
(1+ex
)2
d
dx
e4x
cos(x
2)
( )
−2e4x
sin(x
2 ) −e4x
sin(x
2 )+e4x
cos(x
2 ) −1
2 e4x
sin(x
2 )+ 4e4x
cos(x
2 )
d
dx
1
x3
−2
4
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−3x2
4 (x3
−2)5
4
−4
(x3
−2)5
4
3x2
4 (x3
−2)4
5
d
dx
tan−1
x2
−1
( )
( )
−1
(x2
−1)3
1
x x2
−1
1
x2
x2
−1
!3

4. (Q18.) !
(A) ! (B) ! (C) !
(Q19.) !
(A) ! (B) ! (C) !
(Q20.) Find ! for !
(A) ! (B) ! (C) !
(Q21.) Find ! for !
(A) ! (B) ! (C) !
(Q22.) Find ! for !
(A) ! (B) ! (C) !
(Q23.) Find ! for !
(A) ! (B) ! (C) !
(Q24.) Find ! for !
(A) ! (B) ! (C) !
(Q25.) Find ! for !
(A) ! (B) ! (C) !
d
dx
lnx
x3
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎟
3x2
−lnx
x6
1
3x3
1−3lnx
x4
d
dx
xx
( )
xx
(lnx +1) xx
(lnx −1) xx
(lnx −x)
dy
dx
x3
+ y 3
= 6xy
x2
2x −y 2
x2
−2y
2x −y 2
x2
+ y 2
2x +2y
dy
dx
y siny = x sinx
x cosx
y cosy
sinx −x cosx
siny −y cosy
sinx + x cosx
siny + y cosy
dy
dx
ln
x
y
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
= exy 3
y −xy 4
exy 3
x +3x2
y 3
exy 3
y −x2
y 3
exy 3
x +3x2
y 3
exy 3
y +2x2
y 3
exy 3
x −3x2
y 3
exy 3
dy
dx
x = secy
1
x sinx
1
x x2
−1
1
x2
−1
dy
dx
x −y
( )
2
= sin(x)+sin(y)
cosx +cosy
2(x −y)
2x −2y +cosx
2x −2y −cosy
2x −2y −cosx
2x −2y +cosy
dy
dx
x y
= y x
xy lny −y 2
xy lnx −x2
xy lny −2y
xy lnx −2x
xy lny + y 2
xy lnx + x2
!4

5. (Q26.) Find ! for !
(A) ! (B) ! (C) !
(Q27.) Find ! for !
(A) ! (B) ! (C) !
(Q28.) Find ! for !
(A) ! (B) ! (C) !
(Q29.) Find ! for !
(A) ! (B) ! (C) !
(Q30.) Find ! for !
(A) ! (B) ! (C) !
(Q31.) !
(A) ! (B) ! (C) !
(Q32.) !
(A) ! (B) ! (C) !
(Q33.) !
(A) ! (B) ! (C) !
dy
dx
tan−1
(x2
y)= x + y 3
2xy +1−x4
y 2
3y 2
+3x4
y 2
−2xy
2xy −1−x4
y 2
3y 2
+3x4
y 4
−x2
3xy −1+ x4
y 2
2y 2
−3x4
y 4
+ x2
dy
dx
x2
x2
−y 2
= 3y
−2xy
3(x2
−y 2
)2
2x
2x −2y
2x −6xy
3x2
−9y 2
dy
dx
e
x
y
= x + y 2
ye
x
y
−y 2
xe
x
y
+2y 3
ye
x
y
−3y 2
xe
x
y
+2y 3
xe
x
y
−3y 2
ye
x
y
+2y 3
dy
dx
(x2
+ y 2
−1)3
= y
3x2
(x2
+ y 2
−1)2
1−3y 2
(x2
+ y 2
−1)2
6x(x2
+ y 2
−1)2
1−6y(x2
+ y 2
−1)2
1+6x(x2
+ y 2
−1)2
1−6y(x2
+ y 2
−1)2
d2
y
dx2
9x2
+ y 2
= 9
81x2
−2xy
y 4
18x +2y
y 3
−81
y 3
d2
dx2
1
9 sec(3x)
( )
sec3
(3x)+sec(3x)tan2
(3x) sec3
(3x)+ 3tan3
(3x) 1
9sec(3x) +tan3
(3x)
d2
dx2
x +1
x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
3−2x
4x
5
2
3−x
4x
5
2
3+ x
4x
5
2
d2
dx2
sin−1
x2
( )
( )
2−4x4
(1−x4
)3
2−2x4
(1−x4
)3
2+2x4
(1−x4
)3
!5

6. (Q34.) !
(A) ! (B) ! (C) !
(Q35.) !
(A) ! (B) ! (C) !
(Q36.) !
(A) ! (B) ! (C) !
(Q37.) !
(A) ! (B) ! (C) !
(Q38.) !
(A) ! (B) ! (C) !
(Q39.) !
(A) ! (B) ! (C) !
(Q40.) !
(A) ! (B) ! (C) !
(Q41.) !
(A) ! (B) ! (C) !
d2
dx2
1
1+cosx
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
cosx +cos2
x +2sin2
x
(1+cosx)3
cosx +cos2
x −2sin2
x
(1+cosx)3
1+cosx +2sin2
x
(1+cosx)3
d2
dx2
x tan−1
x
( )
2tan−1
x
x2
+1
2
(x2
+1)2
−2tan−1
x
(x2
+1)2
d2
dx2
x4
ln(x)
( )
7x2
+12x3
lnx 7x +12x2
lnx 7x2
+12x2
lnx
d2
dx2
e−x2
( )
(4x2
−2)e−x2
−2e−x2
(6x2
−2x)e−x2
d2
dx2
cos(lnx)
( )
sin(lnx)+cos(lnx)
x2
sin(lnx)−cos(lnx)
x2
sin( 1
x )−cos( 1
x )
x2
d2
dx2
ln cosx
( )
( )
cscx secx secx tanx −sec2
x
d
dx
1−x2
+ x sin−1
x
( )
sin−1
x
1+ x
2 1−x2
cos−1
x
d
dx
x 4−x2
( )
8−4x2
4−x2
4−2x2
4−x2
8+ x −x2
2 4−x2
!6

7. (Q42.) !
(A) ! (B) ! (C) !
(Q43.) !
(A) ! (B) ! (C) !
(Q44.) !
(A) ! (B) ! (C) !
(Q45.) !
(A) ! (B) ! (C) !
(Q46.) !
(A) ! (B) ! (C) !
(Q47.) !
(A) ! (B) ! (C) !
(Q48.) !
(A) ! (B) ! (C) !
(Q49.) !
(A) ! (B) ! (C) !
d
dx
x2
−1
x
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
−1
(x2
−1)3
1
x x2
−1
1
x2
x2
−1
d
dx
x
x2
−1
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
−1
(x2
−1)3
1
x x2
−1
1
x2
x2
−1
d
dx
cos sin−1
x
( )
( )
−x
1−x2
−x
1−x2
x
1−x2
d
dx
ln x2
+3x +5
( )
( )
2x +3
(x2
+3x +5)2
1
x2
+3x +5
2x +3
x2
+3x +5
d
dx
tan−1
(4x)
( )
2
( )
8tan−1
(4x)
1+16x2
2
1+ tan−1
(4x)
( )
2
8tan−1
(4x)
1+ tan−1
(4x)
( )
2
d
dx
x2
3
( )
2x
3 2
3 x
3
3 x
2
d
dx
sin x lnx
( )
( )
cos
2+lnx
2 x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
2x +lnx
( )cos x lnx
( )
2 x
2+lnx
( )cos x lnx
( )
2 x
d
dx
csc(x2
)
( )
−2x csc(x2
)cot(x2
) −csc(x2
)cot(x2
) −csc(2x)cot(2x)
!7

8. (Q50.) !
(A) ! (B) ! (C) !
(Q51.) !
(A) ! (B) ! (C)!
(Q52.) !
(A) ! (B) ! (C) !
(Q53.) !
(A) ! (B) ! (C) !
(Q54.) !
(A) ! (B) ! (C) !
(Q55.) !
(A) ! (B) ! (C) !
(Q56.) !
(A) ! (B) ! (C) !
(Q57.) !
(A) ! (B) ! (C) !
d
dx
x2
−1
lnx
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2x2
lnx −x2
+1
x(lnx)2
xlnx −x2
+1
2x(lnx)2
2x2
lnx −x2
−1
x(lnx)2
d
dx
10x
( )
10x
ln10
x 10
( )x−1
10x
ln10
d
dx
x +(lnx)2
3
( )
x +2lnx
3x x +(lnx)2
( )
2
3
x +2lnx
3 x +(lnx)2
( )
2
3
1+2lnx
3x x +(lnx)2
3
d
dx
x
3
4
−2x
1
4
( )
3 x −2
4 x3
4
3 x −1
4 x3
4
3 x −2
2 x3
d
dx
log2 x 1+ x2
( )
( )
1+2x
(1+ x2
)ln2
1+ x
x(1+ x2
)ln2
1+2x2
x(1+ x2
)ln2
d
dx
x −1
x2
−x +1
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
−x2
+2x
(x2
−x +1)2
x2
−3x
(x2
−x +1)2
2x2
−x
(x2
−x +1)2
d
dx
1
3 cos3
x −cosx
( )
sin4
x cos2
x sin3
x sin3
x cosx
d
dx
ex cosx
( )
−x sinxex cosx
ex cosx
x sinx −cosx
( ) ex cosx
−x sinx +cosx
( )
!8

9. (Q58.) !
(A) ! (B) ! (C) !
(Q59.) !
(A) ! (B) ! (C) !
(Q60.) !
(A) ! (B) ! (C) !
(Q61.) !
(A) ! (B) ! (C) !
(Q62.) !
(A) ! (B) ! (C) !
(Q63.) !
(A) ! (B) ! (C) !
(Q64.) !
(A) ! (B) ! (C) !
(Q65.) !
(A) ! (B) ! (C) !
d
dx
x − x
( ) x + x
( )
( )
2x −1 x −
1
4x
1−
1
4x
d
dx
cot−1 1
x
( )
( )
−x2
1+ x2
1
1+ x2
−1
1−x2
d
dx
x tan−1
x −ln x2
+1
( )
( )
x2
+1 lnx tan−1
x tan−1
x
d
dx
x 1−x2
2
+
sin−1
x
2
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
1−x2
x sin−1
x sin−1
x 1−x2
d
dx
sinx −cosx
sinx +cosx
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
2
sinx +cosx
2
sinx +cosx
( )2
2sin2
x −2cos2
x
sinx +cosx
( )2
d
dx
4x2
2x3
−5x2
( )
( )
40x4
−80x3
8x5
−20x4
48x3
−80x2
d
dx
x 4−x2
( )
( )
4−5x2
2 x
4−2x
2 x
4−x2
4 x
d
dx
1+ x
1−x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2
(1−x)2
1+ x
1
1+ x (1−x)3
1
2 1−x (1+ x)3
!9

10. (Q66.) !
(A) ! (B) ! (C) !
(Q67.) !
(A) ! (B) ! (C) !
(Q68.) !
(A) ! (B) ! (C) !
(Q69.) !
(A) ! (B) ! (C) !
(Q70.) !
(A) ! (B) ! (C) !
(Q71.) !
(A) ! (B) ! (C) !
(Q72.) !
(A) ! (B) ! (C) !
(Q73.) !
(A) ! (B) ! (C) !
d
dx
sin(sinx)
( )
cosx sin(cosx) cos(cosx) cosx cos sinx
( )
d
dx
1+e2x
1−e2x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
4e2x
(1−e2x
)2
2e2x
(1−e2x
)2
−2e2x
(1−e2x
)2
d
dx
x
1+lnx
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
x
1+lnx
( )2
lnx
1+lnx
( )2
xlnx
1+lnx
( )2
d
dx
x
x
lnx
( )
x
x
lnx
1+lnx
( ) x
x
lnx
x +lnx
( ) x
x
lnx
d
dx
ln
x2
−1
x2
+1
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
2x
x4
−1
x −2
x4
−1
−2x2
x4
−1
d
dx
tan−1
(2x +3)
( )
1
4x2
+12x +10
1
2x2
+6x +5
1
2x2
+6x +3
d
dx
cot4
(2x)
( )
16cot3
(2x)csc2
(2x) 8cot3
(2x)csc2
(2x) −8cot3
(2x)csc2
(2x)
d
dx
x2
1+ 1
x
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2x3
+3x2
(x +1)2
2x3
−3x2
(x +1)2
3x3
−2x2
(x +1)2
!10

11. (Q74.) !
(A) ! (B) ! (C) !
(Q75.) !
(A) ! (B) ! (C) !
(Q76.) !
(A) ! (B) ! (C) !
(Q77.) !
(A) ! (B) ! (C) !
(Q78.) !
(A) ! (B) ! (C) !
(Q79.) !
(A) ! (B) ! (C) !
(Q80.) !
(A) ! (B)! (C) !
(Q81.) !
(A) ! (B)! (C) !
d
dx
e
x
1+x2
( )
e
x
1+x2
1−2x2
( )
1+ x2
( )
2
e
x
1+x2
1−x2
( )
1+ x2
( )
2
e
x
1+x2
−1+2x
( )
1+ x2
( )
2
d
dx
sin−1
x
( )
3
( )
3 cos−1
x
( )
2 3cos−1
x
1−x2
3 sin−1
x
( )
2
1−x2
d
dx
1
2 sec2
(x)−ln(secx)
( )
tan3
x tan2
x secx 2tan2
x secx
d
dx
ln(ln(lnx))
( )
1
ln ln(lnx)
( )
1
xlnx ln lnx
( )
1
x ln lnx
( )
d
dx
π3
( )
4π4
3π2
0
d
dx
ln x + 1+ x2
( )
( )
1
1+ x2
1
x + 1+ x2
2x
x + 1+ x2
d
dx
sinh−1
x
( )
1
x + 1+ x2
1
1+ x2
x2
x + 1+ x2
d
dx
ex
sinhx
( )
2e2x
ex
coshx e2x
!11

12. (Q82.) !
(A) ! (B) ! (C) !
(Q83.) !
(A) ! (B) ! (C) !
(Q84.) !
(A) ! (B) ! (C) !
(Q85.) !
(A) ! (B) ! (C) !
(Q86.) !
(A) ! (B) ! (C) !
(Q87.) !
(A) ! (B) ! (C) !
(Q88.) !
(A) ! (B) ! (C) !
(Q89.) !
(A) ! (B) ! (C) !
(Q90.) !
(A) ! (B) ! (C) !
d
dx
sech 1
x
( )
( )
sech( 1
x )tanh( 1
x )
x2
−sech( 1
x )tanh( 1
x )
x2
1
sech2
( 1
x )
d
dx
cosh lnx
( )
( )
2x2
−1
x2
x2
−1
2x2
x2
+1
2x2
d
dx
ln coshx
( )
( )
cothx −tanhx tanhx
d
dx
sinhx
1+coshx
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
1
1+coshx
1
(1+coshx)2
−1
(1+coshx)2
d
dx
tanh−1
cosx
( )
( )
cschx −cscx −secx
d
dx
x tanh−1
x +ln 1−x2
( )
sech2
x lnx tanh−1
x tanh−1
x
d
dx
sinh−1
tanx
( )
( )
secx sechx coshx
d
dx
sin−1
tanhx
( )
( )
secx sechx coshx
d
dx
tanh−1
x
1−x2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
1−2x tanh−1
x
(1−x2
)2
tanh−1
x
( )
2 1+2x tanh−1
x
(1−x2
)2
!12

13. (Q91.) ! , use the definition of derivative
(Q92.) ! , use the definition of derivative
(Q93.) ! , use the definition of derivative
(Q94.) ! , use the definition of derivative
(Q95.) ! , use the definition of derivative
(Q96.) ! , use the definition of derivative
(Q97.) ! , use the definition of derivative
(Q98.) ! , use the definition of derivative
(Q99.)! , use the definition of derivative
(Q100.) ! , use the definition of derivative
(Q101.) !
d
dx
x3
( )
d
dx
3x +1
( )
d
dx
1
2x +5
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
d
dx
1
x2
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎟
d
dx
sinx
( )
d
dx
secx
( )
d
dx
sin−1
x
( )
d
dx
tan−1
x
( )
d
dx
f (x)g(x)
( )
d
dx
f (x)
g(x)
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
d
dx
3
x
( )
!13