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Differentiation — JEE Main & Advanced Mathematics PYQs

190 previous year questions from Differentiation with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

190 questionsMathematicsSolutions on every page
1

Let R denote the set of all real numbers. Consider the polynomial function f: R R defined by f(x) = d¹⁰ dx¹⁰ ((x^2 - 1)¹⁰ ) , for all x R . Here d¹⁰ dx¹⁰ ((x^2 - 1)¹⁰ ) is the 10th

JEE Advanced 2026 Solution View → 2

Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x R , f'(1) = 2g'(1) = 4 and g(2) = 3f(2) = 9 . Then f(25) - g(25) is equal to :

JEE Main 2026 Solution View → 3

Let f be a real polynomial of degree n such that f(x) = f'(x) f''(x) , for all x R . If f(0) = 0 , then 36 (f'(2) + f''(2) + ₀^2 f(x) ,dx ) is equal to:

JEE Main 2026 Solution View → 4

Let f(x)=x³+x² f^ (1)+2 x f^ (2)+f^ (3), x R . Then the value of f^ (5) is:

JEE Main 2026 Solution View → 5

Let R denote the set of all real numbers. Let f: R R and g: R (0,4) be functions defined by f(x)= _e (x^2+2 x+4 ) , and g(x)= 4 1+e^ -2 x Define the composite function f g⁻¹ by (f

JEE Advanced 2025 Solution View → 6

Let f: R R be a twice differentiable function such that ( x y)(f(2 x+2 y)-f(2 x-2 y))=( x y )(f(2 x +2 y )+f(2 x -2 y )) , for all x , y R . If f^ (0)= 1 2 , then the value of 24 f

JEE Main 2025 Solution View → 7

If _e y=3 ⁻¹ x , then (1-x^2 ) y^ -x y^ at x= 1 2 is equal to

JEE Main 2024 Solution View → 8

Let f(x)=a x^3+b x^2+c x+41 be such that f(1)=40, f^ (1)=2 and f^ (1)=4 . Then a ^2+ b ^2+ c ^2 is equal to:

JEE Main 2024 Solution View → 9

Suppose for a differentiable function h, h(0)=0, h(1)=1 and h^ (0)=h^ (1)=2 . If g (x)=h ( e ^x ) e ^ h(x) , then g^ (0) is equal to:

JEE Main 2024 Solution View → 10

If f(x)= array l x^3 ( 1 x ), x 0 0 , x=0 array . then

JEE Main 2024 Solution View → 11

If y( )= 2 + 2 3 +4 2 +5 +2 , then at = 2 , y^ +y^ +y is equal to :

JEE Main 2024 Solution View → 12

Let f(x)=x^5+2 e ^ x / 4 for all x R . Consider a function g(x) such that (g f)(x)=x for all x R . Then the value of 8 g^ (2) is :

JEE Main 2024 Solution View → 13

If y = x + 1 x 2 − x x x + x + x + 1 15 3 cos 2 x − 5 cos 3 x , then 96 y ' π 6 is equal to:

JEE Main 2024 Solution View → 14

Let y = log e 1 - x 2 1 + x 2 , - 1 < x < 1 . Then at x = 1 2 , the value of 225 y ' - y " is equal to

JEE Main 2024 Solution View → 15

Suppose f x = 2 x + 2 - x tan x tan - 1 x 2 - x + 1 7 x 2 + 3 x + 1 3 . Then the value of f ' 0 is equal to

JEE Main 2024 Solution View → 16

Let f ( x ) = x 3 + x 2 f ' ( 1 ) + x f " ( 2 ) + f ''' ( 3 ) , x ∈ R . Then f ' ( 10 ) is equal to

JEE Main 2024 Solution View → 17

Let for a differentiable function f : ( 0 , ∞ ) → R , f x - f y ≥ log e x y + x - y , ∀ x , y ∈ 0 , ∞ . Then ∑ n = 1 20 f ' 1 n 2 is equal to

JEE Main 2024 Solution View → 18

Let f x = sin x + cos - 2 sin x - cos x , x &#8712; 0 , &#960; - &#960; 4 , then f 7 &#960; 12 f &#34; 7 &#960; 12 is equal to

JEE Main 2023 Solution View → 19

If 2 x y + 3 y x = 20 , then d y d x at 2 , 2 is equal to:

JEE Main 2023 Solution View → 20

If y x = x x , x &#62; 0 , then y &#34; 2 - 2 y &#39; 2 is equal to :

JEE Main 2023 Solution View → 21

Let y = f x = sin 3 &#960; 3 cos &#960; 3 2 - 4 x 3 + 5 x 2 + 1 3 2 . Then, at x = 1 ,

JEE Main 2023 Solution View → 22

Let y x = 1 + x 1 + x 2 1 + x 4 1 + x 8 1 + x 16 . Then y &#39; - y &#34; at x = - 1 is equal to

JEE Main 2023 Solution View → 23

Let x t = 2 2 cos t sin 2 t and y t = 2 2 sin t sin 2 t , t &#8712; 0 , &#960; 2 . Then 1 + d y d x 2 d 2 y dx 2 at t = &#960; 4 is equal to

JEE Main 2022 Solution View → 24

For the curve C : x 2 + y 2 - 3 + x 2 - y 2 - 1 5 = 0 , the value of 3 y &#39; - y 3 y &#39; &#39; , at the point &#945; , &#945; , &#945; &#62; 0 , on C , is equal to ________.

JEE Main 2022 Solution View → 25

The value of log e 2 d dx log cos x cosec x at x = &#960; 4 is

JEE Main 2022 Solution View → 26

If y x = x x x , x &#62; 0 then d 2 x d y 2 + 20 at x = 1 is equal to

JEE Main 2022 Solution View → 27

If cos - 1 y 2 = log e x 5 5 , y &#60; 2 , then

JEE Main 2022 Solution View → 28

Let f : &#8477; &#8594; &#8477; satisfy f x + y = 2 x f y + 4 y ( f x , &#8704; x , y &#8712; &#8477; . If f 2 = 3 , then 14 &#183; f &#39; 4 f &#39; 2 is equal to _____.

JEE Main 2022 Solution View → 29

Let f : R &#8594; R be defined as f x = x 3 + x - 5 . If g x is a function such that f g x = x , &#8704; x &#8712; R , then g &#39; 63 is equal to ______

JEE Main 2022 Solution View → 30

If y = tan - 1 sec x 3 - tan x 3 , &#960; 2 &#60; x 3 &#60; 3 &#960; 2 , then

JEE Main 2022 Solution View → 31

If y ( x ) = cot - 1 1 + sin x + 1 - sin x 1 + sin x - 1 - sin x , x &#8712; &#960; 2 , &#960; , then d y d x at x = 5 &#960; 6 is:

JEE Main 2021 Solution View → 32

If y 1 / 4 + y - 1 / 4 = 2 x , and x 2 - 1 d 2 y d x 2 + &#945; x d y d x + &#946; y = 0 , then | &#945; - &#946; | is equal to _______.

JEE Main 2021 Solution View → 33

Let f ( x ) = cos 2 tan - 1 sin cot - 1 1 - x x , 0 &#60; x &#60; 1 . Then:

JEE Main 2021 Solution View → 34

If y = y ( x ) is an implicit function of x such that log e ( x + y ) = 4 x y , then d 2 y d x 2 at x = 0 is equal to

JEE Main 2021 Solution View → 35

If f x = sin cos - 1 1 - 2 2 x 1 + 2 2 x and its first derivative with respect to x is - b a log e 2 when x = 1 , where a and b are integers, then the minimum value of a 2 - b 2 is

JEE Main 2021 Solution View → 36

Let f : S &#8594; S where S = 0 , &#8734; be a twice differentiable function such that f x + 1 = x f x . If g : S &#8594; R be defined as g x = log e f x , then the value of g &#39

JEE Main 2021 Solution View → 37

The derivative of tan - 1 1 + x 2 - 1 x with respect to tan - 1 2 x 1 - x 2 1 - 2 x 2 at x = 1 2 is :

JEE Main 2020 Solution View → 38

If a + 2 b cos x ) ( a - 2 b cos y = a 2 - b 2 , where a &#62; b &#62; 0 , then d x d y at &#960; 4 , &#960; 4 is:

JEE Main 2020 Solution View → 39

If y 2 + log e cos 2 x = y , x &#8712; - &#960; 2 , &#960; 2 then :

JEE Main 2020 Solution View → 40

If x = 2 sin &#952; - sin 2 &#952; and y = 2 cos &#952; - cos 2 &#952; , &#952; &#8712; 0 , 2 &#960; , then d 2 y d x 2 at &#952; = &#960; is:

JEE Main 2020 Solution View → 41

Let f and g be differentiable functions on R such that f o g is the identity function. If for some a , b &#8712; R , g &#39; a = 5 and g a = b , then f &#39; b is equal to:

JEE Main 2020 Solution View → 42

Let f be any function continuous on a , b and twice differentiable on a , b . If all x &#8712; a , b , f &#39; x &#62; 0 and f &#39; &#39; x &#60; 0 , then for any c &#8712; a , b

JEE Main 2020 Solution View → 43

Let f x = sin tan - 1 x + sin cot - 1 x 2 - 1 , x &#62; 1 . If d y d x = 1 2 d d x sin - 1 f x and y 3 = &#960; 6 , then y - 3 is equal to:

JEE Main 2020 Solution View → 44

Let y = y x be a function of x satisfying y 1 - x 2 = k - x 1 - y 2 where k is a constant and y 1 2 = - 1 4 .Then d y d x at x = 1 2 , is equal to

JEE Main 2020 Solution View → 45

If y &#945; = 2 t a n &#945; + c o t &#945; 1 + t a n 2 &#945; + 1 s i n 2 &#945; , &#945; &#8712; 3 &#960; 4 , &#960; , then d y d &#945; at &#945; = 5 &#960; 6 is

JEE Main 2020 Solution View → 46

Let x k + y k = a k , a , k &#62; 0 and d y d x + y x 1 3 = 0 , then k is

JEE Main 2020 Solution View → 47

If y = log 10 ⁡ x + log x ⁡ 10 + log x ⁡ x + log 10 ⁡ 10 , then d y d x is equal to

NTA Abhyas JEE Main 2020 Solution View → 48

If f x = x + 1 ; x > 1 0 ; x = 1 7 - 3 x ; x < 1 then f ′ 0 equals to

NTA Abhyas JEE Main 2020 Solution View → 49

If log 10 ⁡ x 3 - y 3 x 3 + y 3 = 2 , then d y d x =

NTA Abhyas JEE Main 2020 Solution View → 50

Let ϕ ( x ) be the inverse of the function f ( x ) and f ′ x = 1 1 + x 5 , then d d x ϕ x is equal to

NTA Abhyas JEE Main 2020 Solution View → 51

If y = t a n - 1 s e c x - t a n x , then the value of d y d x is

NTA Abhyas JEE Main 2020 Solution View → 52

The differential coefficient of log 10 ⁡ x with respect to log x 10 is

NTA Abhyas JEE Main 2020 Solution View → 53

If l n x + y = 2 x y , then y ′ 0 is equal to

NTA Abhyas JEE Main 2020 Solution View → 54

Let y = x x x … ∞ , then d y d x is equal to (given x > 0 )

NTA Abhyas JEE Main 2020 Solution View → 55

If 2 y = cot - 1 &#8289; 3 cos &#8289; x + sin &#8289; x cos &#8289; x - 3 sin &#8289; x 2 &#160; &#8704; x &#8712; 0 , &#960; 2 , then d y d x &#160; is equal to

NTA Abhyas JEE Main 2020 Solution View → 56

The rate of change of ( x 2 + 16 ) with respect to x x - 1 at x = 3 is

NTA Abhyas JEE Main 2020 Solution View → 57

If y = t a n - 1 1 x 2 + x + 1 + t a n - 1 1 x 2 + 3 x + 3 + t a n - 1 1 x 2 + 5 x + 7 + t a n - 1 1 x 2 + 7 x + 13 , x > 0 and d y d x x = 0 = - k 1 + k , then the value of k is

NTA Abhyas JEE Main 2020 Solution View → 58

If f x = cos &#8289; x . cos &#8289; 2 x . cos &#8289; 4 x &#160; . cos &#8289; 8 x &#160; . cos &#8289; 16 x , then f ' &#960; 4 is ( where &#8201;&#8201; f ' ( x ) = d dx f ( x )

NTA Abhyas JEE Main 2020 Solution View → 59

If x = 1 - t 2 1 + t 2 and y = 2 t 1 + t 2 , t h e n d y d x is equal to

NTA Abhyas JEE Main 2020 Solution View → 60

If u = x 2 + y 2 and x = s + 3 t , &#160; y = 2 s - t , then d 2 u d s 2 is equal to (where t is a constant)

NTA Abhyas JEE Main 2020 Solution View → 61

If y = a cos &#8289; ( log &#8289; x ) + b sin &#8289; ( log &#8289; x ) , where a &#160; &#38; &#160; b are parameters, then x 2 y &#34; + x y &#39; is equal to

NTA Abhyas JEE Main 2020 Solution View → 62

If y = l n x x , then d 2 y d x 2 = 2 x 3 l n x - &#955; , where &#955; is

NTA Abhyas JEE Main 2020 Solution View → 63

If f x = 2 + x 1 + x 1 + x , then f ′ 0 is equal to

NTA Abhyas JEE Main 2020 Solution View → 64

If f x = x 3 + 3 x + 1 and g x is the inverse function of f x , then the value of g ' 5 is equal to

NTA Abhyas JEE Main 2020 Solution View → 65

If y = 1 x , then the value of d y 1 + y 4 + d x 1 + x 4 + 3 is equal to

NTA Abhyas JEE Main 2020 Solution View → 66

If y = t a n - 1 x 1 + 6 x 2 + t a n - 1 2 x - 1 2 x + 1 ∀ x > 0 , then d y d x is equal to

NTA Abhyas JEE Main 2020 Solution View → 67

If x = sec &#8289; t + tan &#8289; t and y = sec &#8289; t - tan &#8289; t , where t is a parameter, then the value of d y d x when x = 1 3 is

NTA Abhyas JEE Main 2020 Solution View → 68

If y = 1 + x y + s i n - 1 s i n 2 x , then d y d x at x = 0 is

NTA Abhyas JEE Main 2020 Solution View → 69

If f ( x ) = c o s - 1 s i n x 4 c o s 2 x - 1 , then 1 &#960; f ' &#960; 3 &#8901; f &#960; 10 is

NTA Abhyas JEE Main 2020 Solution View → 70

If f x is a twice differentiable function such that f " x = - f x , f ' x = g x , h x = f 2 x + g 2 x and h 10 = 10 , then h 5 is equal to

NTA Abhyas JEE Main 2020 Solution View → 71

Let g x = x - 2 and h x = g g x be two functions, then the value of h ' - 1 + h ' 1 + h ' 3 + h ' 5 is equal to (where, h ' denotes the derivative of h )

NTA Abhyas JEE Main 2020 Solution View → 72

If x = 3 cos ⁡ t and y = 5 sin ⁡ t , where t is a parameter, then 9 d 2 y d x 2 at t = - π 6 is equal to

NTA Abhyas JEE Main 2020 Solution View → 73

If y = cos ⁡ x cos ⁡ 2 x cos ⁡ 4 x cos ⁡ 8 x , then d y d x at x = π 2 is

NTA Abhyas JEE Main 2020 Solution View → 74

If y = tan ⁡ x - sin ⁡ x , then the value of d y d x at x = 5 π 4 is

NTA Abhyas JEE Main 2020 Solution View → 75

Let f x = x 2 - 4 x - 3 , x > 2 and g ( x ) be the inverse of f ( x ) . Then the value of 1 g ' ( 2 ) , where f ( x ) = 2 , is (here, g ' represents the first derivative of g )

NTA Abhyas JEE Main 2020 Solution View → 76

If f x = 2 sin &#8289; x - 1 - 2 cot &#8289; x , then the value of f ' &#960; 3 is equal to

NTA Abhyas JEE Main 2020 Solution View → 77

If f : R &#8594; R is a function defined as f x 3 = x 5 , &#160; &#8704; x &#8712; R - 0 and f x is differentiable &#8704; x &#8712; R , then the value of 1 4 f ' 27 is equal to (h

NTA Abhyas JEE Main 2020 Solution View → 78

If y = 2 + sin ⁡ x + 2 + sin ⁡ x + 2 + sin ⁡ x + … ∞ , then the value of d y d x at x = 0 is

NTA Abhyas JEE Main 2020 Solution View → 79

Let f x is a differentiable function on x &#8712; R , such that f x + y = f x f y for all x , y &#8712; R where f 0 &#8800; 0 . If f 5 = 10 , &#160; f ' 0 = 6 , then the value of f

NTA Abhyas JEE Main 2020 Solution View → 80

If f x = x - 4 x - 4 + t a n - 1 1 - 2 x 2 + x , &#160; &#8704; 4 &#60; x &#60; 8 , then the value of f &#8242; 5 is equal to

NTA Abhyas JEE Main 2020 Solution View → 81

If f x = x &#8211; 1 x &#8211; 2 x &#8211; 3 x &#8211; 4 x &#8211; 5 , then the value of f &#8242; 5 is equal to

NTA Abhyas JEE Main 2020 Solution View → 82

Consider the function f x = t a n - 1 3 x - 2 3 + 2 x , ∀ x ≥ 0 . If g x is the inverse function of f x , then the value of g ′ π 4 is equal to

NTA Abhyas JEE Main 2020 Solution View → 83

Let f x = t a n - 1 x 3 - 1 x 2 + x , then the value of 17 f &#8242; 2 is equal to

NTA Abhyas JEE Main 2020 Solution View → 84

If I n = d n d x n x n ln &#8289; x , then the value of 1 50 I 7 - 7 I 6 is equal to

NTA Abhyas JEE Main 2020 Solution View → 85

Let y = x l o g e x . If the value of d y d x at x = e 4 is k , then the value of 4 e 3 k is (use e = 2.7 )

NTA Abhyas JEE Main 2020 Solution View → 86

If x y . y x = 16 , then the value of d y d x at ( 2 , 2 ) is

NTA Abhyas JEE Main 2020 Solution View → 87

Let f x be a differentiable function on x ∈ R such that f x + y = f x ⋅ f y for all x , y . If f 0 ≠ 0 , f 5 = 12 and f ′ 0 = 16 , then f ′ 5 is equal to

NTA Abhyas JEE Main 2020 Solution View → 88

If y = s i n - 1 2 x 1 + x 2 , then d y d x is

NTA Abhyas JEE Main 2020 Solution View → 89

If the function f x = cos - 1 ⁡ x 3 2 - 1 - x - x 2 + x 3 (where, ∀ 0 < x < 1 ), then the value of 3 f ' 1 2 is equal to (take 3 = 1.73 )

NTA Abhyas JEE Main 2020 Solution View → 90

Consider a function f x = x x , ∀ x ∈ 1 , ∞ . If g x is the inverse function of f x , then the value of g ′ 4 is equal to

NTA Abhyas JEE Main 2020 Solution View → 91

If ϕ x = log 8 ⁡ log 3 ⁡ x , then ϕ ′ e is equal to

NTA Abhyas JEE Main 2020 Solution View → 92

If f x = s i n - 1 2 &#215; 3 x 1 + 9 x , then f ' - 1 2 is equal to

NTA Abhyas JEE Main 2020 Solution View → 93

The derivative of t a n - 1 s i n x - c o s x s i n x + c o s x with respect to x 2 , where x &#8712; 0 , &#960; 2 , is

JEE Main 2019 Solution View → 94

If e y + x y = e , the ordered pair d y d x , d 2 y d x 2 at x = 0 is equal to

JEE Main 2019 Solution View → 95

If f 1 = 1 , f &#39; 1 = 3 , then the derivative of f f f x + f x 2 at x = 1 is:

JEE Main 2019 Solution View → 96

If 2 y = cot - 1 3 cos x + sin x cos x - 3 sin x 2 &#8704; x &#8712; 0 , &#960; 2 , then d y d x is equal to

JEE Main 2019 Solution View → 97

Let f be a differentiable function such that f 1 = 2 and f ' x = f x for all x &#8712; R . If h x = f f x , then h ' 1 is equal to :

JEE Main 2019 Solution View → 98

For x &#62; 1 , if 2 x 2 y = 4 e 2 x - 2 y , then 1 + log e &#8289; 2 x 2 d y d x is equal to

JEE Main 2019 Solution View → 99

If x _ e ( _ e x )-x²+y²=4(y>0), then d y d x at x=e is equal to :

JEE Main 2019 Solution View → 100

Let, f : R &#8594; R be a function such that f x = x 3 + x 2 f ' 1 + x f '' 2 + f ''' 3 , &#8704; x &#8712; R . Then f 2 equals

JEE Main 2019 Solution View → 101

If x = 3 t a n t and y = 3 s e c t , then the value of d 2 y d x 2 at t = &#960; 4 , is:

JEE Main 2019 Solution View → 102

If x = 2 cosec - 1 t and y = 2 sec - 1 t , t &#8805; 1 , then d y d x is equal to

JEE Main 2018 Solution View → 103

If f(x)= ⁻¹ ( 2 3^x 1+9^x ) , then f^ (- 1 2 ) equals.

JEE Main 2018 Solution View → 104

If x^2+y^2+ y=4 , then the value of d^2 y d x^2 at the point (-2,0) is

JEE Main 2018 Solution View → 105

If x 2 + y 2 + sin y = 4 , then the value of d 2 y d x 2 at the point - 2 , 0 is :

JEE Main 2018 Solution View → 106

If 2 x = y 1 5 + y - 1 5 and x 2 - 1 d 2 y d x 2 + &#955; x d y d x + k y = 0 , then &#955; + k is equal to

JEE Main 2017 Solution View → 107

If y = x + x 2 - 1 15 + x - x 2 - 1 15 , then x 2 - 1 d 2 y d x 2 + x d y d x is equal to

JEE Main 2017 Solution View → 108

If for x &#8712; 0 , 1 4 , the derivative of tan - 1 &#8289; 6 x x 1 - 9 x 3 is x &#8901; g x , then g x equals:

JEE Main 2017 Solution View → 109

Let f : R &#8594; R , g : R &#8594; R and h : R &#8594; R be differentiable functions such that f x = x 3 + 3 x + 2 , g f x = x and h g g x = x , for all x &#8712; R . Then,

JEE Advanced 2016 Solution View → 110

Let f and g be two differentiable functions on R such that f^ (x)>0 and g^ (x) < 0 for all x R . Then for all x :

JEE Main 2014 Solution View → 111

If y = e n x , then d 2 y d x 2 . d 2 x d y 2 is equal to :

JEE Main 2014 Solution View → 112

Let f &#8289; x = x sin &#960; x , x &#62; 0 . Then for all natural numbers n , f &#8289; &#8242; x vanishes at

JEE Advanced 2013 Solution View → 113

If f(x)= ( x) and f^ (x)+ x f^ (x)+g(x)=0 , then g(x) is :

JEE Main 2013 Solution View → 114

For a>0, t (0, 2 ) , let x= a^ ⁻¹ t and y= a^ ⁻¹ t , Then, 1+ ( d y d x )^2 equals :

JEE Main 2013 Solution View → 115

Let f(x)= x^2-x x^2+2 x x 0,-2 . Then d d x [f⁻¹(x) ] (wherever it is defined) is equal to:

JEE Main 2013 Solution View → 116

If y = sec tan - 1 x , then d y d x at x = 1 is equal to

JEE Main 2013 Solution View → 117

If f^ (x)= ( x) and y=f ( 2 x+3 3-2 x ) , then d y d x equals

JEE Main 2012 Solution View → 118

d^2 x d y^2 equals

JEE Main 2011 Solution View → 119

Let f be a real-valued function defined on the interval (-1,1) such that e^ -x f(x)=2+ ₀^x t^4+1 d t, for all x (-1,1) and let f⁻¹ be the inverse function of f . Then (f⁻¹ )^ (2) i

JEE Advanced 2010 Solution View → 120

Let f:(-1,1) R be a differentiable function with f(0)=-1 and f^ (0)=1 . Let g(x)=[f(2 f(x)+2)]^2 . Then g^ (0)=

JEE Main 2010 Solution View →
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