# Differentiation · Mathematics · JEE Main

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JEE Main 2024 (Online) 9th April Evening Shift
If $$\log _e y=3 \sin ^{-1} x$$, then $$(1-x^2) y^{\prime \prime}-x y^{\prime}$$ at $$x=\frac{1}{2}$$ is equal to
JEE Main 2024 (Online) 9th April Morning Shift
Let $$f(x)=a x^3+b x^2+c x+41$$ be such that $$f(1)=40, f^{\prime}(1)=2$$ and $$f^{\prime \prime}(1)=4$$. Then $$a^2+b^2+c^2$$ is equal to:
JEE Main 2024 (Online) 30th January Evening Shift
Let $$f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$$ be a function satisfying $$f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$$ for all $$x, y, f(y) \neq ... JEE Main 2024 (Online) 30th January Morning Shift Let$$g: \mathbf{R} \rightarrow \mathbf{R}$$be a non constant twice differentiable function such that$$\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\...
JEE Main 2024 (Online) 30th January Morning Shift
If $$f(x)=\left|\begin{array}{ccc} 2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4... JEE Main 2024 (Online) 29th January Evening Shift$$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 ...
JEE Main 2024 (Online) 29th January Morning Shift
Suppose $$f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}$$. Then the value of $$f^{\prim... JEE Main 2023 (Online) 13th April Morning Shift For the differentiable function$$f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$$, let$$3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10$$, then$$\lef...
JEE Main 2023 (Online) 8th April Morning Shift
Let $$f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$$. Then $$f\left(\frac{7 \pi}{12}\right) f^{\prime... JEE Main 2023 (Online) 6th April Morning Shift If$$2 x^{y}+3 y^{x}=20$$, then$$\frac{d y}{d x}$$at$$(2,2)$$is equal to : JEE Main 2023 (Online) 1st February Evening Shift If$$y(x)=x^{x},x > 0$$, then$$y''(2)-2y'(2)$$is equal to JEE Main 2023 (Online) 1st February Morning Shift Let$$f(x) = 2x + {\tan ^{ - 1}}x$$and$$g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$$. Then JEE Main 2023 (Online) 31st January Morning Shift Let$$y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$$. ... JEE Main 2023 (Online) 29th January Evening Shift Let$$f$$and$$g$$be the twice differentiable functions on$$\mathbb{R}$$such that$$f''(x)=g''(x)+6xf'(1)=4g'(1)-3=9f(2)=3g(2)=12$$. The... JEE Main 2023 (Online) 25th January Morning Shift Let$$y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})$$. Then$$y' - y''$$at$$x = - 1$$is equal to JEE Main 2023 (Online) 24th January Evening Shift If$$f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$$, then JEE Main 2022 (Online) 28th July Evening Shift Let$$x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}$$and$$y(t)=2 \sqrt{2} \sin t \sqrt{\sin 2 t}, t \in\left(0, \frac{\pi}{2}\right)$$. Then$$\frac{1+\lef...
JEE Main 2022 (Online) 26th July Evening Shift
The value of $$\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$$ at $$x=\frac{\pi}{4}$$ is
JEE Main 2022 (Online) 27th June Morning Shift
If $${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| JEE Main 2022 (Online) 25th June Morning Shift Let f : R$$\to$$R be defined as$$f(x) = {x^3} + x - 5$$. If g(x) is a function such that$$f(g(x)) = x,\forall 'x' \in R$$, then g'(63) is equal to... JEE Main 2022 (Online) 24th June Evening Shift If$$y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2}
JEE Main 2021 (Online) 27th August Evening Shift
If $$y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left... JEE Main 2021 (Online) 26th August Morning Shift Let$$f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$$, 0 < x < 1. Then : JEE Main 2020 (Online) 5th September Evening Slot The derivative of$${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$$with respect to$${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}}...
JEE Main 2020 (Online) 4th September Morning Slot
If $$\left( {a + \sqrt 2 b\cos x} \right)\left( {a - \sqrt 2 b\cos y} \right) = {a^2} - {b^2}$$ where a > b > 0, then $${{dx} \over {dy}}\,\,at... JEE Main 2020 (Online) 3rd September Morning Slot If y2 + loge (cos2x) = y,$$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$, then : JEE Main 2020 (Online) 9th January Evening Slot If$$x = 2\sin \theta - \sin 2\theta $$and$$y = 2\cos \theta - \cos 2\theta $$,$$\theta \in \left[ {0,2\pi } \right]$$, then$${{{d^2}y} \over {...
JEE Main 2020 (Online) 9th January Evening Slot
Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b $$\in$$ R, g'(a) = 5 and g(a) = b, then ƒ'(b) ...
JEE Main 2020 (Online) 8th January Morning Slot
Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1. If $${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \... JEE Main 2020 (Online) 7th January Evening Slot Let y = y(x) be a function of x satisfying$$y\sqrt {1 - {x^2}} = k - x\sqrt {1 - {y^2}} $$where k is a constant and$$y\left( {{1 \over 2}} \right...
JEE Main 2020 (Online) 7th January Morning Slot
Let xk + yk = ak, (a, k > 0 ) and $${{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0$$, then k is:...
JEE Main 2020 (Online) 7th January Morning Slot
If $$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }... JEE Main 2019 (Online) 12th April Evening Slot The derivative of$${\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)$$, with respect to$${x \over 2}$$, where$$\left( {x ...
JEE Main 2019 (Online) 12th April Morning Slot
If ey + xy = e, the ordered pair $$\left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right)$$ at x = 0 is equal to :
JEE Main 2019 (Online) 10th April Evening Slot
Let f(x) = loge(sin x), (0 < x < $$\pi$$) and g(x) = sin–1 (e–x ), (x $$\ge$$ 0). If $$\alpha$$ is a positive real number such that a = (fog...
JEE Main 2019 (Online) 8th April Evening Slot
If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is :
JEE Main 2019 (Online) 8th April Morning Slot
If $$2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}$$, x $$\in$$ $$\left( {... JEE Main 2019 (Online) 12th January Morning Slot For x > 1, if (2x)2y = 4e2x$$-$$2y, then (1 + loge 2x)2$${{dy} \over {dx}}$$is equal to : ... JEE Main 2019 (Online) 11th January Morning Slot If xloge(logex)$$-$$x2 + y2 = 4(y > 0), then$${{dy} \over {dx}}$$at x = e is equal to : JEE Main 2019 (Online) 10th January Morning Slot Let f : R$$ \to $$R be a function such that f(x) = x3 + x2f'(1) + xf''(2) + f'''(3), x$$ \in $$R. Then f(2) equals - JEE Main 2019 (Online) 9th January Evening Slot If x$$=$$3 tan t and y$$=$$3 sec t, then the value of$${{{d^2}y} \over {d{x^2}}}$$at t$$ = {\pi \over 4},$$is : JEE Main 2018 (Online) 16th April Morning Slot If$$x = \sqrt {{2^{\cos e{c^{ - 1}}}}} $$and$$y = \sqrt {{2^{se{c^{ - 1}}t}}} \,\,\left( {\left| t \right| \ge 1} \right),$$then$${{dy} \over {dx...
JEE Main 2018 (Online) 15th April Evening Slot
If    f(x) = sin-1 $$\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right),$$ then f'$$\left( { - {1 \over 2}} \right)$$ equals :
JEE Main 2018 (Online) 15th April Morning Slot
If $$f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \... JEE Main 2018 (Online) 15th April Morning Slot If x2 + y2 + sin y = 4, then the value of$${{{d^2}y} \over {d{x^2}}}$$at the point ($$-$$2,0) is : JEE Main 2017 (Online) 9th April Morning Slot Let f be a polynomial function such that f (3x) = f ' (x) . f '' (x), for all x$$ \in $$R. Then : JEE Main 2017 (Online) 8th April Morning Slot If y =$${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$$then (x2$$-$$1)$${{{d^2}y} \over {d{x^...
JEE Main 2017 (Offline)
If for $$x \in \left( {0,{1 \over 4}} \right)$$, the derivatives of $${\tan ^{ - 1}}\left( {{{6x\sqrt x } \over {1 - 9{x^3}}}} \right)$$ is $$\sqrt x ... JEE Main 2014 (Offline) If$$g$$is the inverse of a function$$f$$and$$f'\left( x \right) = {1 \over {1 + {x^5}}},$$then$$g'\left( x \right)$$is equal to: JEE Main 2013 (Offline) If$$y = \sec \left( {{{\tan }^{ - 1}}x} \right),$$then$${{{dy} \over {dx}}}$$at$$x=1$$is equal to : AIEEE 2011$${{{d^2}x} \over {d{y^2}}}$$equals: AIEEE 2010 Let$$f:\left( { - 1,1} \right) \to R$$be a differentiable function with$$f\left( 0 \right) = - 1$$and$$f'\left( 0 \right) = 1$$. Let$$g\left( ...
AIEEE 2009
Let $$y$$ be an implicit function of $$x$$ defined by $${x^{2x}} - 2{x^x}\cot \,y - 1 = 0$$. Then $$y'(1)$$ equals
AIEEE 2006
If $${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$ then $${{{dy} \over {dx}}}$$ is
AIEEE 2004
If $$x = {e^{y + {e^y} + {e^{y + .....\infty }}}}$$ , $$x > 0,$$ then $${{{dy} \over {dx}}}$$ is
AIEEE 2003
If $$f\left( x \right) = {x^n},$$ then the value of $$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{... AIEEE 2003 Let$$f\left( x \right)$$be a polynomial function of second degree. If$$f\left( 1 \right) = f\left( { - 1} \right)$$and$$a,b,c$$are in$$A.P, $$... AIEEE 2002 If$$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},$$then$$\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$$is ## Numerical JEE Main 2024 (Online) 4th April Evening Shift Let$$f: \mathbb{R} \rightarrow \mathbb{R}$$be a thrice differentiable function such that$$f(0)=0, f(1)=1, f(2)=-1, f(3)=2$$and$$f(4)=-2$$. Then, ... JEE Main 2024 (Online) 1st February Evening Shift If y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x, then 96 y^{\prime}\left(... JEE Main 2024 (Online) 27th January Morning Shift Let f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}. Then f^{\prime}(10) is equal to ____________. JEE Main 2023 (Online) 13th April Evening Shift Let$$f(x)=\sum_\limits{k=1}^{10} k x^{k}, x \in \mathbb{R}$$. If$$2 f(2)+f^{\prime}(2)=119(2)^{\mathrm{n}}+1$$then$$\mathrm{n}$$is equal to _____... JEE Main 2023 (Online) 1st February Morning Shift If$$f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)$$and$$g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)$$, then the value of$$f(4)-g(4)$$i... JEE Main 2023 (Online) 30th January Morning Shift Let$$f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}$$For$$\mathrm{n} \geq 2$$, define$$f^{\mathrm{n}}(x)=f^{1} \mathrm...
JEE Main 2023 (Online) 29th January Morning Shift
Let $$f:\mathbb{R}\to\mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}$$. If $$f'(0)... JEE Main 2022 (Online) 27th July Evening Shift For the curve$$C:\left(x^{2}+y^{2}-3\right)+\left(x^{2}-y^{2}-1\right)^{5}=0$$, the value of$$3 y^{\prime}-y^{3} y^{\prime \prime}$$, at the point ... JEE Main 2022 (Online) 29th June Evening Shift Let f and g be twice differentiable even functions on ($$-$$2, 2) such that$$f\left( {{1 \over 4}} \right) = 0$$,$$f\left( {{1 \over 2}} \right) = 0...
JEE Main 2022 (Online) 27th June Evening Shift
If $$y(x) = {\left( {{x^x}} \right)^x},\,x > 0$$, then $${{{d^2}x} \over {d{y^2}}} + 20$$ at x = 1 is equal to ____________.
JEE Main 2022 (Online) 26th June Evening Shift
Let f : R $$\to$$ R satisfy $$f(x + y) = {2^x}f(y) + {4^y}f(x)$$, $$\forall$$x, y $$\in$$ R. If f(2) = 3, then $$14.\,{{f'(4)} \over {f'(2)}}$$ is equ...
JEE Main 2021 (Online) 26th August Morning Shift
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then $${{{d^2}y} \over {d{x^2}}}$$ at x = 0 is equal to ___________.
JEE Main 2021 (Online) 17th March Morning Shift
If $$f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)$$ and its first derivative with respect to x ...
JEE Main 2020 (Online) 2nd September Evening Slot
If y = $$\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}}$$, then $${{dy} \over {dx}}$$ at x = 0 i...
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