CLASS 12 DIFFERENTIATION DPP
CONTINUITY AND DIFFERENTIATION
1. State the function which is continuous for all x ∈ R.
(a) sin x (b) (x²-25)/(x-5)
(c) [x] (d) sgn (x)
2. The function 'f' defined by f(x) = (x³-8)/(x-2), x≠2 is 12, x=2
(a) not continuous at x = 2
(b) continuous at x = 2
(c) not continuous at x = 3
(d) not continuous at x = - 2
3. If x = at², y = 2at, then d²y/dx² is
(a) 1/t (b) -1/t³
(c) at² (d) -1/(2at³)
4. Derivative of (x/2)√(a²-x²) + (a²/2)sin⁻¹(x/a), with respect to x, is
(a) sin⁻¹(x/a) (b) (x/2)√(a²-x²)
(c) √(a²-x²) (d) 1/√(a²-x²)
5. Discuss the applicability of LMV Theorem for the function
f(x) = |x|, in [-1, 1]
(a) applicable, c = 0 (b) applicable, c = -1
(c) applicable, c = 1 (d) not applicable
Differentiate each of the following with respect to x
(Exercises 6 to 10):
6. sin log x 7. cos⁻¹√x
8. e⁻¹/x 9. sin[log (x²-1)]
10. logₑ(sin x)
11. Differentiate cos x with respect to eˣ.
12. If y = sec⁻¹(√(x+1) / √(x-1)) + sin⁻¹(√(x-1) / √(x+1)), find dy/dx
13. Given f(0) = - 2, f'(0) = 3. Find h'(0), where h(x) = xf(x).
14. Find dy/dx at (4, 9), when √x+√y = 5.
15. Find the second derivative of log x with respect to x.
16. Show that the function f(x) = sin x / |x| is discontinuous at x = 0.
17. Show that the function f(x) = |x| - x| is continuous at x = 0
18. Let f be the function defined as
f(x) = { 2x/[√(a+x) - √(a-x)], if x ≠ 0 ; 3k, if x = 0} , a > 0.
For what value of k, function is continuous at x = 0?
19. A function f is defined as
f(x) = { (1-cos 4x) / x², if x < 0; x / (√16+√x - 4), if x > 0}
Is the function continuous at x = 0? If not how should the function be
defined at x = 0, so that the function is continuous at x = 0?
20. If a function f defined as
f(x) = { sin(a+1)x + sin x / x², if x < 0; c, if x = 0; x / (x+bx² - x), if x > 0}
is continuous at x = 0. Find the values of a, b and c.
21. Let f be the function defined as
f(x) = { (cos²x - sin²x - 1) / √(x+1) - 1, if x ≠ 0 ; k, if x = 0 }
is continuous at x = 0. Find the value of k.
22. Let f be the function defined as
f(x) = log(1 + 3x) - log(1 - 5x) / 2x , if x ≠ 0 k , if x = 0
is continuous at x = 0. Find the value of k.
23. Show that the function f(x) =
x / |x| + x² , if x ≠ 0 k , if x = 0
is discontinuous at x = 0 whatever may be the value of k.
24. Examine the continuity of the function
f(x) = |x| cos 1/x , if x ≠ 0 0 , if x = 0
at x = 0.
25. Let f be the function defined as f(x) = e^(1/x-1) , if x ≠ 0
0 , if x = 0
Show that function is discontinuous at x = 0.
26. Let f be the function defined as
f(x) = x² + 2 - 16 / x⁴ - 16 , if x ≠ 2
k , if x = 2
be continuous at x = 2.
Find the value of k.
27. For what value of k, the function f(x) = sin 3x / tan 2x , if x < 0
k , if x = 0 is continuous at x = 0?
log(1 + 3x) / e^x , if x > 0
28. If function f is differentiable at x = a, find
lim x→a x²f(a) - a²f(x) / x - a
29. Let f(x) = x + k , if x ≤ 1 show that f is continuous
kx² + 1 , if x > 1 at x = 1. Find k, so that f is differentiable at x = 1
Differentiate each of the following with respect to x in Exercises 30 to 49:
30. sin⁻¹ (a + bcos x / b + a cos x)
31. log (x + √(a² + x²))
32. sin (m sin⁻¹ x)
33. logₑ (sin x)
34. √(a² + x²) + √(a² - x²) / √(a² + x²) - √(a² - x²)
35. cos⁻¹ (√(1 + cos 2x) / 2)
36. tan⁻¹ (a cos x - b sin x / b cos x + a sin x)
37. cot⁻¹ (√(1 + x³ - x))
38. sin⁻¹ (√(x - 1) - √(1 - x²))
39. sin⁻¹ (√(1 + x) + √(1 - x) / 2)
40. tan⁻¹ (2a / 1 - a²) , a > 1
41. x² - 2x
42. xˣ
43. log(xˣ + cosec x)
44. x(cos x) + (cos x)ˣ
45. sin⁻¹ (5x + 12 √(1 - x²) / 13)
46. tan⁻¹ (x / √(1 - x²) + √(1 + x²) / 1 - x² )
47. log (√(x² + a²) - x / √(x² + a²) + x)
48. (x/y)ˣ
49. √(x - 3) (x² + 4) / 3x² + 4x + 5
50. If y = x^y + e^x, prove that dy/dx = √y² - 4
51. If y = log (√(1 + x) - √(x - 1)), show that dy/dx = 1 / 2√(x² - 1)
52. If ax² + 2hxy + by² + 2gx + 2fy + c = 0, show that
dy/dx = -(ax + hy + g) / (hx + by + f)
53. Find dy/dx, if sin(xy) + x/y = x² - y
54. If x² + y² = r - 1 and x⁴ + y⁴ = r² + 1/r², show that x² dy/dx / y = 1
55. If y = x sin y, show that dy/dx = y / x(1 - x cos y)
56. If xy = (x + y)ⁿ, x ≠ m y, show that dy/dx = y / x
57. Given that cos(x/2) cos(x/4) cos(x/8) = sin x / sin x,
prove that 1/2² + 1/2² * sec² (x/4) + 1/2⁴ * sec² (x/8) + ... =
cosec x - 1/x
58. If y = e^(x+y), show that dy/dx = y / 1 - y
59. If e^x = y^x, show that dy/dx = (log y)² / log y - 1
60. If eˣ + e - x = c, show that dy/dx = -eˣ
61. If x = a cos³ θ and y = a sin³ θ, find dy/dx at θ = π/4
62. Differentiate tan⁻¹( (2x / 1 - x²)) with respect to sin⁻¹ ((2x / 1 + x²))
63. Differentiate tan⁻¹ (√(1 + x²) - √(1 - x²) / √(1 + x²) + √(1 - x²)) with
respect to cos⁻¹ x².
64. Prove that the derivative of tan⁻¹(√(1+x²)-1)/x with respect to tan⁻¹x
is independent of x.
65. If ey(x+1) ~ 1, show that (d²y/dx²) = (dy/dx)²
66. If log y = tan⁻¹x, show that (1 + x²)y₂ + (2x - 1)y₁ = 0.
67. If x = a sec θ and y = a tan θ, find d²y/dx² at θ = π/4
68. If y = xx, prove that xyy₂ - xy₁² - y² = 0.
69. If x = a(θ + sin θ) and y = a(1 - cos θ), find d²y/dx² at θ = π/2
70. If y = x log (x/(a+bx)), prove that d²y/dx² = 1/x (a/(a+bx))²
71. If f(x) = ((3+x)/(1+x))^(2+3x), find f'(0).
72. If y = tan⁻¹ (5ax/(a²-6x²)).
Prove that dy/dx = 3a/(a² + 9x²) + 2a/(a² + 4x²)
73. If y√(1 + x²) = log[√(1+x²) - x].
Show that (x² + 1)dy/dx + xy + 1 = 0.
74. Verify Rolle's Theorem for the function
f(x) = log(x² + 2) - log 3 on [-1, 1].
State which of the following is continuous as well as Long Answer I / Long Answer II Type
1. differentiable for x ∈ R
(a) |x| (b) [x]
(c) polynomial function (d) sgn (x)
2. Derivative of with respect to x, is
3. Find the relationship between a and b so that the function f defined by
f(x)= ax + 1, if x ≤ 3 bx + 3, if x > 3
is continuous at x = 3.
[NCERT: AI 2011]
(a) 2 (b) 1/(x-1)2
(c) (2x-1) / (x-1)2 (d) -1/(x-1)2
4. For what value of λ is the function defined by
λ(x²-2x), if x ≤ 0
f(x)= 4x + 1, if x> 0 continuous at x = 0?
[NCERT: Foreign 2011]
5. Derivative of x/x-1 with respect to x, is
(a) 2 (b) 1/(x-1)2
(c) 2x-1 / (x-1)2 (d) -1/(x-1)2
6. Derivative of sin x with respect to log x, is
(a) x/ cos x (b) cos x / x
(c) x cos x (d) x² cos x
7. Differentation for x C R
(a) |x| (b) [x]
(c) polynomial function (d) sgn[x]
8. Examine the continuity of the function f(x)= x+3' x ∈ R.
9. State the points of discontinuity for the function
f(x)= [x], in -3 < x < 3.
[HOTS]
10. Find the point of discontinuity if any for the function
f(x)= 1/x-5
[NCERT]
11. Differentiate y = e* + e²* + e³* + e** + e5* with respect to x.
12. If y= 500e³x + 600e-5x, show that dy = 49y. [NCERT]
13. Verify the Rolle's Theorem for the function f(x)= sin x in [0, π].
14. Verify the Rolle's Theorem for the function f(x)= x in [-1, 1].
15. Verify Mean Value Theorem for the function
f(x)=(x-1)2/³ in [0, 2].
[HOTS]
16. Find the derivative of f(log x) with respect to x, where
f(x) = log x.
17. Find the derivative of sin⁻¹x/1+sin⁻¹x with with respect to sin⁻¹x.
18. The derivative of a differentiable even function is odd function. State
true or false.
19. It is known that for x ≠ 1, 1 + x + x² + ... + x"-¹ = 1-xn/1-x
Hence find the sum of the series 1 + 2x + 3x² + ... + (n-1) x¹-2
22. Show that the function f(x) = 2x - |x| is continuous but
not differentiable at x = 0.
[Foreign 2013]
23. Is the function f(x)=3x+4tanx / x continuous x=0? If not how should
we define the function to make it continuous?
24. If f(x) =if x ≠ 0, find whether f(x) is continuous at x = 0.
[HOTS]
25. Is the function f(x)=x-1 if x ≠1 continuous
F(x) = {sin5x/3x x!=0
{ k x=0
26. For what value of k is the following function continuous at x2= 0?
f(x)sin 5x x ≠ 0
27. Discuss the continuity of the function
log(1+3x)
28. Show that the function defined by f(x) = cos x² is a continuous function.
[NCERT]
29. Find the value of k such that the function
2x² +2 -16 if x ≠ 2
f(x)=4-16 is continuous at x = 2.
k, if x = 2
[NCERT Exemplar]
30. For what value of k, is the following function continuous at x = 0?
f(x)= { 1-cos4x / 8x2 if x!= 0
{ k if x = 0
[NCERT Exemplar]
Differentiate each of the following with respect to x
(Exercises 31 to 40):
31. y = sin-¹ (5x+12√1-x²/13)
32. y = √(sec x -1)/(sec x + 1)
33. y = cos⁻¹ ((3x + 4√(1-x²))/5)
34. f(x) = tan⁻¹((1-x)/(1+x)) - tan⁻¹((x + 2)/(1-2x))
35. y = sin⁻¹(cos x) + cos⁻¹(sin x). [HOTS]
36. y = sin⁻¹(√(1+x) + √(1-x) )/2
37. y = sin⁻¹ [2a√(1 - a²x²)]
38. y = cot⁻¹((1+x)/(1-x))
39. y = tan⁻¹(√(1+a²x² - 1)/(ax)) [HOTS]
40. y = tan⁻¹(√(1+x²) - √(1-x²)) / (√(1+x²) + √(1-x²)) , x² ≤ 1.
[Delhi 2015]
41. Differentiate cot⁻¹(√(1+sinx) + √(1-sinx)) / (√(1+sinx) - √(1-sinx)), 0 < x < π/2 [NCERT]
42. Differentiate sec⁻¹(1/(4x³ - 3x)), 0 < x < 1/√2 w.r.t. x
[NCERT Exemplar]
43. Prove that d/dx [ x/2√(a² - x²) + a²/2 sin⁻¹ x/a] = √(a² - x²)
[Foreign 2011]
44. Differentiate tan⁻¹(√(1-x²)/x) with respect to cos⁻¹(2x√(1-x²)),
when x ≠ 0. [Delhi 2014]
45. Differentiate tan⁻¹(√(1+x²) - 1)/x with respect to sin⁻¹((2x)/(1+x²)),
when x ≠ 0. [Delhi 2014]
46. Find dy/dx, if y = tan⁻¹(x / (1 + √(1-x²))
47. If x = cos t(3 - 2cos² t) and y = sin t(3 - 2sin² t), find the value of dy/dx
at t = π/4. [AI 2014]
48. If x = a cos θ + b sin θ and y = a sin θ - b cos θ, show that y² d²y/dx² - x
dy/dx + y = 0. [Delhi 2015, Foreign 2011]
49. If eᶻ + eᵞ = eˣ, prove that dy/dx + eᶻ = 0. [Foreign 2014]
50. If f(x) = √(x² + 1); g(x) = (x+1)/(x² + 1) and h(x) = 2x - 3, then find
f'{h'[g(x)]}. [Foreign 2015]
51. If y = (x + √(1+x²)), then show that
(1+x²) *d²y/dx² + x * dy/dx = n²y. [Foreign 2010]
52. Find dy/dx, if y = cos(log x²). [HOTS]
53. Find dy/dx, if y = cos x + (sin x)¹/x
54. Find dy/dx, if y = x² - cos x + x² - 1/x² + 1
55. If x = a(cos t + log tan t/2), y = a(1 + sin t), find d²y/dx²
56. If x = a(θ – sin θ), y = a(1 + cos θ), find d²y/dx² . [Delhi 2011]
57. If x = a(cos t + t sin t) and y = a(sin t – t cos t),
0 < t < π/2, find d²y/dx², dy/dx and d²y/dx² .
[Delhi 2017(C), AI 2012]
58. If x = a sin t and y = a(cos t + log tan t/2), find d²y/dx² .
[Delhi 2013]
59. If x = a cos³ θ and y = a sin³ θ, then find the value of d²y/dx² at
θ = π/6. [AI 2013]
60. If x log y + y log x = 5, find dy/dx.
61. If log (x² + y²) = 2tan⁻¹ (y/x), then show that dy/dx = x+y/x-y
[Delhi 2019]
62. If y = log (x + √(x² + 1)), then prove that
(x² + 1) * d²y/dx² + x * dy/dx = 0. [Delhi 2013; Foreign 2011]
63. If y = sin⁻¹ x, show that (1 - x²) * d²y/dx² - x * dy/dx = 0.
[NCERT; Delhi 2012]
64. If y = 3 cos(log x) + 4 sin(log x), show that
x² * d²y/dx² + x * dy/dx + y = 0 [NCERT; AI 2016; Delhi 2012]
65. If yˣ = e^(y - x), prove that dy/dx = (1 + log y)²/log y . [AI 2013]
66. If xʸ = e^(x-y), prove that dy/dx = log x/(1 + log x)² .
[NCERT Exemplar; AI 2013]
67. Differentiate, tan⁻¹ (√(1+x²)-1)/x) with respect to tan⁻¹ x, when
x ≠ 0. [NCERT Exemplar; Foreign 2013]
68. If y = sin⁻¹(x²√(1-x²) + x√(1-x⁴)), then prove that
dy/dx = 2x/√(1-x⁴) + 1/√(1-x²)
69. If √(1-x⁸) + √(1-y⁶) = a³(x³ - y³), prove that
dy/dx = x²y⁴/y²x⁶
70. If x² = e^(e^(-y)), show that
dy/dx = 2 - log x/(1 - log x)²
71. If sin y = xcos (a + y), show that
dy/dx = cos²(a + y)/cos a
Also, show that dy/dx = cos a, when x=0. [Delhi 2018(C)]
72. If the derivative of tan⁻¹(a + bx) takes the value 1 at
x = 0, prove that b = 1 + a².
73. If y = e^(e^(e^(-x))), prove that
dy/dx = y/1 - y
74. Differentiate tan⁻¹(2x/1-x²) with respect to sin⁻¹(2x/1+x²)
75. If y = f((2x-1)/(x²+1)) and f'(x) = sin λ², find dy/dx. [HOTS]
76. If y=sin(π sin⁻¹ x), prove that (1-x²)y₂-xy₁+π²y=0.
77. Differentiate tan⁻¹((3x-x³)/(1-3x²)) w.r.t. tan⁻¹(x/√(1-x²))
78. If y = √(1 - sin 2x)/(1 + sin 2x), show that dy/dx + sec²(π/4 - x) = 0.
79. If x² + y² = 2 (or a or b or a + b in place of 2), find dy/dx.
80. Find dy/dx, if tan (x + y) + tan (x - y) = 1. [HOTS]
81. If x²y = 1, find dy/dx.
82. If x sin (a + y) + sin a cos (a + y) = 0, prove that
dy/dx = sin²(a + y)/sin a [NCERT Exemplar]
83. If x = e^(x/y), prove that dy/dx = x - y/x log x [NCERT]
84. If √(1-x²) + √(1-y²) = a(x - y), prove that
dy/dx = √(1 - y²)/√(1 - x²) [NCERT Exemplar, HOTS]
85. If cos y = x cos (a + y) with cos a ≠ 1, prove that
dy/dx = cos²(a + y)/sin a [NCERT; Foreign 2014]
86. Verify the Rolle's Theorem for the function
f(x) = sin⁴x + cos⁴x in [0, π/2]
87. Verify the Rolle's Theorem for the function
f(x) = sin 3x in [0, π].
88. Verify the Rolle's Theorem for the function
f(x) = √4 - x² in [-2, 2]. [NCERT Exemplar]
89. Verify Mean Value Theorem for the function
f(x) = √25 - x² in [-3, 4].
90. If f(x) and g(x) are functions derivable in [a, b] such
that f(a) = 4, f(b) = 10, g(a) = 1, g(b) = 3, show that
for a < c < b, we have f'(c) = 3g'(c). [HOTS]
91. Examine the following function f(x) for continuity at
x = 1 and differentiability at x = 2.
f(x) = { 5x-4 , 0 < x < 1
{4x² - 3x , 1 < x < 2
{3x + 4 , x ≥ 2 [Guwahati 2015]
92. If y = x³ log(1/x), then prove that x²(d²y/dx²) - 2(dy/dx) + 3x² = 0.
[Guwahati 2015]
93. If (x / x-y) = log( a / x-y), then prove that dy/dx = 2 - (x / y).
[Guwahati 2015]
94. Let f(x) = x - |x - x²|, x ∈ [-1, 1]. Find the point of
discontinuity, (if any), of this function in [-1, 1].
[Bhubaneshwar 2015]
95. If y = log((x / (a + bx))^(1/2)), prove that x²(d²y/dx²) =
((dy/dx) - y)².
[Bhubaneshwar 2015]
96. Find the derivative of sec⁻¹((1 / (2x² - 1))) w.r.t. √1 - x² at
x = 1/2. [Bhubaneshwar 2015]
97. If x = α sin 2t (1 + cos 2t) and y = β cos 2t (1 - cos 2t),
show that dy/dx = (β / α) tan t. [Patna 2015]
98. Find d/dx cos⁻¹((x - x⁻¹) / (x + x⁻¹)). [Patna 2015]
99. Find the derivative of the following function f(x) w.r.t.
x, at x = 1: cos⁻¹(sin(√(1 + x) / 2) + x²) [Patna 2015]
100. If function f(x) = |x - 3| + |x - 4|, then show that f(x) is
not differentiable at x = 3 and x = 4. [Chennai 2015]
101. If y = x² / 2, find dy/dx. [Chennai 2015]
102. If y = √x+1 - √x-1, prove that
(x² - 1)(d²y/dx²) + x(dy/dx) - 1/4y = 0 [Chennai 2015]
* * * * * * * * * * * * * * * *
