DIFFERENTIATION REVISION SHEET 1

Q.1. If f ( x)  x  1 , find
d
( fof )( x) Q.10. If x  aet (sin t  cos t ) and y  aet (sin t  cos t ) , then
dx
dy x  y
Ans. 1 prove that  .
dx x  y
Q.2. If f ( x)  x  7 and g ( x)  x  7, x  R , then find
dy
Q.11. If (sin x) y  x  y , find
d dx
( fog)( x) .
dx 1  y( x  y) cot x
Ans.
Ans. 1 ( x  y) log(sin x)  1
dy Q.12. If y  (sec1 x)2 , x  0 , show that
Q.3. If y  x | x | , find for x  0 . Ans.  2 x
dx
d2y dy
dy x 2 ( x 2  1)  ( 2 x3  x)  2  0 .
Q.4. If y  log(cos e ) , then find
x
. dx2
dx
dx
Q.13. If x  sin t , y  sin pt, prove that
Ans.  e x tan e x
dy x  y d2y dy
1 
Q.5. If log( x  y )  2 tan   , show that  (1  x 2 )  x  p2 y  0 .
2 2 y
. dx 2
dx
 x dx x  y
1  cos x 
dy Q.14. Differentiate tan1  with respect to x.
Q.6. If x y  y x  ab , find .  sin x 
dx
1
dy y x log y  y x y 1 Ans. 
Ans.   2
dx x y log x  x y x 1
dy
Q.7. If y  (sin 1 x)2 , prove that Q.15. If ( x2  y 2 )2  xy , find .
dx
d2y dy dy y  4 x ( x 2  y 2 )
(1  x 2 ) 2
x 20 Ans.  
dx dx dx 4 y ( x 2  y 2 )  x

Q.8. If x  cos t  log tan , y  sin t , then find the

t dy
 2
Q.16. If x  a(2  sin 2 ) and y  a(1  cos 2 ) , find
dx
d2y d2y   1
values of 2
and 2
at t  . when   . Ans.
dt dx 4 3 3
Ans. 2 2 Q.17. If y  sin(sin x) , prove that
Q.9. If ( x  a)2  ( y  b)2  c 2 , for some c  0 , prove that d2y dy
2
 tan x  y cos2 x  0
3/ 2 dx dx
  dy  2 
1     d2y 
  dx   Q.18. If x  a sec3  and y  a tan3  , find at   .
is a constant independent of a and b. dx 2 3
d2y
dx2 1 cos4 
Ans.  
Ans. c, (c  0) . 3a tan
1 d2y dy
Q.19. If y  e tan x
, prove that (1  x 2 ) 2
 (2 x  1) 0
dx dx
.
 DIFFERENTIATION REVISION SHEET 2

Q.20. If y  sin 1(6 x 1  9 x2 ) , 

1
x
1  1  x2  1 
, then Q.31. Differentiate tan1  w.r.t. sin 1 2 x ,
3 2 3 2  x  1  x2
 
dy 6
find . Ans. 1
dx 1  9 x2 if x  (1, 1) . Ans.
4
Q.21. Differentiate the function (sin x) x  sin 1 x with Q.32. If x  sin t and y  sin pt , prove that
respect to x .
d2y dy
(1  x 2 )  x  p2 y  0
dy 1 dx 2
dx
Ans.  (sin x) x ( x cot x  log sin x) 
dx 2 x  x2  1  x2  1  x2 
Q.33. If y  tan1 , x 2  1 then find
2
d y  1  x2  1  x2 
Q.22. If x m y n  ( x  y)m  n , prove that  0.  
dx2
dy dy x
dy . Ans. 
Q.23. If x  y  a , then find
y x b
. dx dx 1  x4
dx
dy  [ y x log y  yx y 1 ] Q.34. If x  a cos  b sin  , y  a sin   b cos , show that
Ans. 
dx [ x y log x  xy x 1 ] d2y dy
y2 2
x  y0.
2 dx dx
d 2 y  dy 
Q.24. If e y ( x  1)  1 , then show that   . 1
dx2  dx  Q.35. If y  em sin x , then
dy y ( x  1) dy
Q.25. If xy  e( x  y ) , then show that  . show that (1  x 2 )
dy dy
 x  x  m2 y  0 .
dx x( y  1) dx2
dx dx

Q.26. If f ( x)  sin 2 x  cos 2 x , find f '   . Q.36. If y  ( x  1  x2 )n , then show that
6
d2y dy
Ans.  1 3 (1  x 2 ) 2
x  n2 y .
dx dx

Q.27. If y  (cos x) x  sin 1 3x , find

dy
. Q.37. Find whether the following function is differentiable
dx
at x  1 and x  2 or not :
3
Ans.  (cos x) x { x tan x  log(cos x)}   x, x 1
2 3x 1  3x 
f ( x)   2  x, 1 x  2
d 2 y 1  dy 
2  2  3 x  x 2 , x2
Q.28. If y  x x , prove that
y
     0. 
dx 2
y  dx  x
 
1  1  x
2
Q.38. Differentiate tan  with respect to
Q.29. Differentiate x sin x
 (sin x) cos x
with respect to x .  x 
 
Ans.  xsin x  sin x  cos x log x 
 x  cos1 (2 x 1  x2 ) , when x  0 .
 (sin x)cos x {cos x  cot x  sin x log(sin x)} 1
Ans. 
Q.30. If y  2 cos(log x)  3sin(log x) , prove that 2
2
d2y  dy  y
x
d2y
2 dy
 x  y 0.
Q.39. If y  x x prove that 1     0
 dx 
2
2 dx x
dx dx
 DIFFERENTIATION REVISION SHEET 3

Q.40. If y  Peax  Qebx , show that dy log(cos y)  y tan x

Ans. 
dx log(cos x)  x tan y
d2y dy
2
 (a  b)  aby  0 . 1 1 dy  y
dx dx Q.51. If x  asin t , y  a cos t
, show that  .
dx x
Q.41. If x  a cos  b sin  and y  a sin   b cos , show
Q.52. If x 1  y  y 1  x  0 ,  1  x  1 , then prove
d2y dy
that y 2
x  y0.
dx2
dx dy 1
that, 
dx (1  x)2
dy
Q.42. If e x  e y  e x  y , prove that  eyx  0 . 1
dx Q.53. If y  ea cos x ,  1  x  1 , show that
Q.43. If cos y  x cos(a  y), where cos a  1 , prove that
d2y dy
(1  x 2 ) 2
 x  a2 y  0
dy cos2 (a  y ) dx dx
 .
dx sin a dy 2 y
Q.54. If x16 y9  ( x2  y)17 , prove that  .
Q.44. Differentiate the following function with respect to dx x

x : (log x) x  xlog x . d x 2 a 2 1 x 
Q.55. Prove that  a  x  sin
2
 a x
2 2
dx  2 2 2
 1  log x  2 
Ans.  (log x) x log(log x)    x    log x 
 log x  x  Q.56. If y  log( x  x2  1) , then prove that

Q.45. If y  log[ x  x2  a2 ] , show that ( x 2  1)

d2y
x
dy
0.
2
dx dx
d2y dy
( x2  a2 ) x 0. dy
, if y  sin 1[ x 1  x  x 1  x2 ] .
2
dx dx Q.57. Find
dx
dy (1  log y ) 2
Q.46. If y x  e y  x , prove that  . 1 1
dx log y Ans.  y '  
1  x2 2 x 1 x
Q.47. If x  a cos  and y  a sin  , then find the value
3 3
 3x  4 1  x 2 
2
 Q.58. If y  cos1  , then find dy .
d y
at   .
32  5  dx
of 2
Ans.  
dx 6 27 a
Q.48. If x sin (a  y)  sin a cos(a  y)  0, prove that 1
Ans y ' 
1  x2
dy sin 2 (a  y )
 .
 x  dy
dx sin a Q.59. If y  log tan   , show that  sec x .
 4 2 dx
Q.49. Differentiate the following with respect to
 1  sin x  1  sin x 
d2y 
1  Also find the value of at x  .
x : tan  , 0 x . dx 2
4
 1  sin x  1  sin x  2
 t
1 Q.60. If x  a cos t  log tan  , y  a(1  sin t ) , find
Ans.   2
2
d2y 1
dy . Ans. sin t  sec4 t .
Q.50. If (cos x)  (cos y) , then find
y x
. dx 2 a
dx
Q 61. Differentiate the following functions with
 DIFFERENTIATION REVISION SHEET 4

respect to x: 1 1 1 1
(i)  x (ii) x  (iii) x  
(i) sin (sin x), x  [0, 2]
-1
3 3 3 3
(ii) cos-1 (cos x), x  [0, 2] 3 3 3
ANS. (i) (ii) (iii)
1 x2 1 x2 1 x2
Q 62. Differentiate sin 1 (2x 1  x 2 ) with respect to
Q 68. Differentiate each of the following functions
1 1
x, if (i)  x with respect to x
2 2
 1 x2 
1 1 (i) cos1  2 
,0  x  1
 x  1 (iii) 1  x  
(ii)
2 2  1 x 
2 2 2  1 x2 
ANS.. (i) (ii)  (iii)  (ii) sin 1  2 
,0  x 1
1 x2 1 x2 1 x2  1 x 
Q 63. Differentiate sin-1 (3x – 4x3) with respect to 2 2
ANS. (i) (ii) 
1
x, if (i)   x 
1 1
(ii)  x  1 1 x 2
1 x2
2 2 2
Q 69. If y 1  x 2  x 1  y 2  1 , prove that
1
(iii) 1  x  
2 dy 1  y2
 .
3 3 3 dx 1 x2
ANS. (i) (ii)  (iii) 
1 x2 1 x2 1 x2 1  x 2 (2x  3)1/ 2 dy
Q 70. If y  , find
Q 64. Differentiate cos-1 (2x2 – 1) with respect to x, (x  2)
2 2/3
dx
if ANS.
(i) 0 < x < 1 (ii) -1 < x < 0
1  x 2 (2x  3)1/ 2  x 1 4x 
2 2     
ANS.. (i)  (ii) (x 2  2)2/ 3  1  x 2x  3 3(x  2) 
2 2

1 x2 1 x2
x (x  4)3/ 2
Q 65. Differentiate cos 1 (1  2x 2 ) with respect to x, Q 71. Find the derivative of w.r.t x.
(4x  3) 4 / 3
if
ANS.
(i) 0 < x < 1 (ii) -1 < x < 0

ANS. (i)
2
(ii) 
2 x (x  4)3/ 2  1 3 16 
4/3 
  
1 x2 1 x2 (4x  3)  2x 2(x  4) 3(4x  3) 
Q 66. Differentiate cos-1 (4x3 – 3x) with respect to
Q 72. If 1  x 2  1  y 2  a(x  y), prove that
x,if
dy 1  y2
 1 1 1   1 
(i) x    ,  (ii) x   ,1 (iii) x   1,   dx 1 x2
 2 2 2   2
3 3 3 Q 73. If y 1  x 2  x 1  y 2  1 , prove that
ANS.. (i) (ii)  (iii)
1 x2 1 x2 1 x2 dy 1  y2
 .
 3x  x 3  dx 1 x2
Q 67. Differentiate tan 1  2 
, if
 1  3x 
 DIFFERENTIATION REVISION SHEET 5

x x x
Q 74. Given that cos .cos .cos ..... 
sin x  x 2  y2 
, Q 83. If cos1  2 2 
 tan 1 a , prove that
x y 
2 4 8 x
prove that
dy y
1 x 1 x 1 
2
sec 2  4 sec 2  ....  cos ec 2 x  2 dx x
2 2 2 4 x
Q 84. If 1  x 6  1  y 6  a(x 3  y3 ) , prove that
Q 75. If y  sin x  sin x  sin x  .....to  ,
dy x 2 1  y6
 , where -1 < x < 1 and -1 < y < 1.
prove that
dy cos x
 dx y 2 1  x 6
dx 2y  1
1 1
ax
.......  Q 85. If x 2  y 2  t  and x 4  y 4  t 2  2 , then
Q 76. If y  a x , prove that t t
dy 1
dy y2 log y prove that  3 .
 dx x y
dx x(1  y log x.log y)
x e x .........to 
Q 77. If y  e x  e , show that v
x )....... 
Q 78. If y  ( x )( x )(
, show that
dy y2

dx x(2  y log x)
1
Q 79. If y  x  , prove that
1
x
1
x
x  ....
dy y

dx 2y  x
sin x
Q 80. If y  , prove that
cos x
1
sin x
1
cos x
1
1  .......to 
dy (1  y) cos x  ysin x

dx 1  2y  cos x  sin x
y
Q 81. If log(x2 + y2) = 2 tan 1   , show that
x
dy x  y

dx x  y
Q 82. If x 1  y  y 1  x  0 and x  y, prove that
dy 1

dx (x  1) 2