CLASS – XII

DIFFERENTIATION 1

CBSE sample paper 2017

Q4. If y  log x  log x  log x  ..... prove that
Two marks questions
2 y  1 dy  1 .
dx x
dy
Q1. If e y x  1  1 , show that  e y .
dx
 2 x 1 
Q5. Differentiate: sin 1  x 
.
dy 1  4 
Q2. Find if x 3  x 2 y  xy 2  y 3  81 .
dx
dy
Q6. Find if y  x log x loglog x  .
5x 1 1 dx
Q3. If y  tan 1 , x , then prove
1  6x 2
6 6
𝑑2 𝑦
dy 2 3 Q7. If 𝑥 = 𝑠𝑖𝑛𝑡 and 𝑦 = 𝑠𝑖𝑛𝑝𝑡, then show that(1 − 𝑥 2 ) 𝑑𝑥 2 −
that   .
dx 1  4 x 2 1  9 x 2 𝑑𝑦
𝑥 𝑑𝑥 + 𝑝2 𝑦 = 0.

dy sin ax  b 
Q4. Find if y  . d  1 1  x  1
dx coscx  d  Q8. Prove that cos  .

dx  
2  2 1 x 2
dy
Q5. Find when x  a  sin  , y  a1  cos  . 4 marks questions
dx
(2017)
dy
Q6. Find if sin 2 y  cos xy   .
dx Q1. Differentiate the function sin x  x  sin 1 x with respect
f  x  g  x  h x  to x. (2015)
Q7. If y  l m n prove that
d2y
a b c Q2. If x m y n  x  y m n , prove that  0.
dx 2
f  x  g  x  h x 
dy
 l m n (2016)
dx
a b c
Q3. If x  a sin 2t 1  cos 2t  and y  b cos 2t 1  cos 2t  ,
dy  sin x  dy  
Q8. Find y  tan 1   find the value of at t  and t  .
dx  1  cos x  dx 4 3

2
Four marks questions d 2 y 1  dy  y
Q4. If y  x , prove that 2      0 .
x
dx y  dx  x
dy 2  1 x 
Q1. Find , if y  e sin x 2 tan 1 .
dx 
 1  x 
 Q5. Differentiate xsin x  sin x cos x with respect to x.

 1 x 2  1 x 2  Q6. If y  2 coslog x   3sinlog x , prove that

Q2. If y  tan 1  , find dy .
 2  dx d2y dy
 1 x  1 x 
2
x2 2
x y0. (2015)
dx dx
Q3. If x  a cos   b sin  , y  a sin   b cos  , show
 1  x2  1  x2 
d2ydy Q7. If y  tan1 ' x 2  1, then find dy
that y 2
x  y o.  2  dx
 1 x  1 x 
2 2
dx dx
. (2008)

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 CLASS – XII
DIFFERENTIATION 2

1
Q9. If y  e m sin x dy
, then show that Q24. If e x  e y  e x y , prove that  e y x  0
dx
1  x  ddx y  x dy
2
2
2
m y 0. 2
dy

log x dy ex e y 1
 y x
 
 
dx or or, .
dx 1  log x 2
dx e e 1
x cos 1 x
Q10. If y   log 1  x 2 , then prove that dy y
1 x 2
Q25. If x m y n  ( x  y) m n , prove that  . (2012)
dx x
dy cos 1 x
 .

dx 1  x 2 3 2  Q27. If x  cost (3  2 cos2 t ) and y  sin t (3  2 sin 2 t ),
dy 
find the value of at t  .
d y 2
 dx 4
Q11. If x  a sec3  , y  a tan 3  , find at   .
dx 2 4
Q28. If x  a sin 2t (1  cos 2t ) and y  b cos 2t (1  cos 2t ), .
Q12. If x  ae sin t  cost  and y  ae sin t  cost ,
t t
  dy  b
show that at t  ,  
dy x  y 4  dx  a
prove that  .
dx x  y
 1 x2 1
Q13. If y  Aemx  Benx , show that Q29. Differentiate tan 1   with respect to
 x 
 
d2y
 m  n   mny  0
dy
 2x 
dx 2
dx sin 1  , when x  0 .
1 x 2 
x 1
Q14. If f x   x 2  1 ; g ( x)  and hx   2 x  3 , then dy 
x2  1 Q31. Find the value of at   , if
dx 4
find f hg x .
x  ae sin   cos  and y  ae sin   cos  .
(2014)
 x xa
Q32. If y  tan 1    log , prove that
Q18. If y  sin 1  x 1  x  x 1  x 2  a xa
 
dy 2a 3
dy  4 .
and 0  x  1, then find . (2010) dx x  a 4
dx
(2013)
dy
Q19. If (tan 1 x) y  y cot x  1, then find .
dx
t d2y
Q33. If x  a sin t and y  a(cos t  log tan ) find .
Q20. If x  a(cost  t sin t ) and y  a(sint  t cost ), then 2 dx 2

d2y  Q35. Differentiate the following function with

find the value of 2
at t  .
dx 4
respect to x: (log x) x  x log x
x
x y dy Q36. Differentiate the following function w.r.t.x:
Q21. If ( x  y) . e  a, prove that y  x  2 y.
dx  2 x 1.3 x 
sin 1  
 1  36  x 
Q23. If cos y  x cos(a  y), where cos a  1,  

dy cos 2 (a  y)
prove that  .
dx sin a

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 CLASS – XII
DIFFERENTIATION 3

Q37. If x  a cos3  and y  a sin 3  , then find the Q48. If y  (tan1 x) 2

d2y  d2y dy
value of 2
at   . show that ( x 2  1) 2 2
 2 x( x 2  1)  2.
dx 6 dx dx

1
 x  y  e a cos x
,  1  x  1,
Q38. If y  x log   , then prove that Q49. If
 a  bx 
2
d2y  dy  d2y dy
x3  x  y . show that (1  x 2 ) x  a 2 y  0.
dx 2
 dx  dx 2 dx

Q39. If x  2 cos  cos 2 and y  2 sin   sin 2 , thenQ50. If x 1  y  y 1  x  0,  1  x  1, x  y, .

dy  3  dy 1
prove that:  tan  . then prove that 
dx  2  dx (1  x) 2

dy
Q40. If y  (sin x) x  sin 1 x , then find . (2009) Q51. If x  a sin1 t , y  a cos1 t , show that dy   y .
dx dx x

Q41. If y  log  x  x 2  a 2  , then prove that  

1  x 2  1
  Q52. Differentiate tan 1  with respect to x.
 x 
x  ddx y  x dy  
2
2
 a2 2
 0.
dx
Q53. If x  a(cos t  t sin t ) and y  a(sin t  t cos t )
m
 d2y d2y
Q42. If y   x  x 2  1 , then show that 0t  , find and .
  2 dt 2 dx 2

x  1 ddx y  x dy
2
2
m y 0. 2

  ddx y  x dy
2 2
dx Q55. If y  sin 1 x , show that 1  x 2 2
0.
dx
1 
 
2
d y
Q44. If x = tan  log y  , then show that 1  x 2 + (2011)
a  dx 2

2 x  a  dy
=0 (2011) x2 1
dx Q56. Differentiate: x x cos x  w.r.t. x .
x2 1
Q45. If x  cos and y  sin 3  , then prove that
d2y
Q57. If x  a(  sin  ), y  a(1  cos ), find
 
d y  dy 
2 2 .
y     3 sin 2  5 cos 2   1 . dx 2
dx 2  dx 
dy log x
Q58. If x y  e x  y , show that  .
(2012) dx log( xe)2

dy
Q46. If (cos x) y  (cos y) x , find . (2009) d x a2  x 
dx Q59. Prove  a2  x2  sin 1    a 2  x 2
dx  2 2  a 
dy sin 2 (a  y)
Q47. If sin y  x sin(a  y) prove that  .
d2y
Q60. If y  log  x  x 2  1, prove that ( x 2  1) 2  x  0.
dx sin a dy
  dx dx
(2009)
1  tan x dy
Q61. If y  log , prove that  sec 2 x .
1  tan x dx

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 CLASS – XII
DIFFERENTIATION 4

(2010) Q76. Differentiate the following with respect to x:

 1  sin x  1  sin x 
y  cot 1  
, if y  cos x x  sin x 1 x
dy
Q62. Find  1  sin x  1  sin x 
dx  .

   
3x  4 1  x 2
Q64. If y  cos 1   dy 5x  12 1  x 2
 , find dx . Q77. If y  sin 1   , find dy .
 5  
  
13

dx

Q65. If y  cos ec 1x, x  1, then show that 

Q78. If x  a(cos   log tan ) and y  a sin  , find the
  ddx y  2x  1 dydx  0 . 2
2
x x2 1 2
2 dy 
value of at  
dx 4
 
2x  3 1  x 2 
Q66. If y  cos 1 
dy (2007)
 , find
 13  dx
 
Q79. Find the derivative of cos(2 x  1) w.r.t. x from
(2009) first principle.

dy
Q67. Find if ( x 2  y 2 ) 2  xy. 2 mark questions
dx

Q68. If y  3 cos(log x)  4 sin(log x) then show that (2017)

2
d y dy
x2. x  y0
Q81. If y  sin 1  6 x 1  9 x 2 , 
1 1
dx 2 dx x , then
  3 2 3 2
sin 1 x
 d
2 dy
y dy
Q70. If y  , show that 1  x 2 2
 3x  y  0 . find dx .
1 x2 dx dx

1 mark questions
d2y dy
Q71. If y  3e 2x
 2e 3x
, prove that 5  6y  0 .
dx 2 dx (2014)

Q72. If y  e x sin x  cos x , then show that Q82. Write the derivative of Sin x w .r .t cosx.

d2y dy Some other questions

2
2  2y  0 .
dx dx
Q1. Differentiate with respect to x from first principle:
x
e
(2008) Q2. Differentiate with respect to x:
1 1  dy  x 
Q73. If y  x 2  1  log  1  2 
, find . (a) log tan  
x x  dx  4 2
(b) log x  a 2  x 2 
sec x  1    
Q74. If f ( x)  , find f ' ( x). Also find f '  .
sec 1 2 1  cos x
(c) log
1  cos x

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 CLASS – XII
DIFFERENTIATION 5


 1  sin x  1  sin x 

(d) tan 1  , 0  x  

 1  sin x  1  sin x 

1 x 
(e) cot 1  
1  x 
2x 2  3
(f) x cot x 
x2  x  2
(g) y  sin x tan x  cos x sec x

n
Q3.If y  x  a 2  x 2  , then prove that
dy ny
 .
  dx a  x2
2

Q4.If y 
1 x
1 x

, prove that 1  x 2
dy
dx
y0.

Q6. If y  x  tan x , then show that

d2y
cos 2 x  2 y  2x  0 .
dx 2

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