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## Differentiation

### Mathematics

#### Previous Years Questions

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Let $$x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}$$ and $$y(t)=2 \sqrt{2} \sin t \sqrt{\sin 2 t}, t \in\left(0, \frac{\pi}{2}... JEE Main 2022 (Online) 28th July Evening Shift The value of$$\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$$at$$x=\frac{\pi}{4}$$is... JEE Main 2022 (Online) 26th July Evening Shift If$$y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2} ...
JEE Main 2022 (Online) 24th June Evening Shift
If $${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| JEE Main 2022 (Online) 27th June Morning Shift Let f : R$$\to$$R be a function defined by f(x) = (x$$-$$3)n1 (x$$-$$5)n2, n1, n2$$\in$$N. Then, which of the fo... JEE Main 2022 (Online) 29th June Evening Shift The function$$f(x) = {x^3} - 6{x^2} + ax + b$$is such that$$f(2) = f(4) = 0$$. Consider two statements :Statement 1 :... JEE Main 2021 (Online) 1st September Evening Shift If$$y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \...
JEE Main 2021 (Online) 27th August Evening Shift
The local maximum value of the function $$f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$$, x &gt; 0, is
JEE Main 2021 (Online) 26th August Evening Shift
Let $$A = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 &amp; , &amp; {if\,i = j} ... JEE Main 2021 (Online) 20th July Morning Shift If Rolle's theorem holds for the function$$f(x) = {x^3} - a{x^2} + bx - 4$$,$$x \in [1,2]$$with$$f'\left( {{4 \over ...
JEE Main 2021 (Online) 25th February Morning Slot
Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $$\ne$$ 0 for all x $$\... JEE Main 2021 (Online) 24th February Evening Slot The derivative of$${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$$with respect to$${\tan ^{ - 1}...
JEE Main 2020 (Online) 5th September Evening Slot
If the function $$f\left( x \right) = \left\{ {\matrix{ {{k_1}{{\left( {x - \pi } \right)}^2} - 1,} &amp; {x \le \pi ... JEE Main 2020 (Online) 5th September Morning Slot If$$\left( {a + \sqrt 2 b\cos x} \right)\left( {a - \sqrt 2 b\cos y} \right) = {a^2} - {b^2}$$where a &gt; b &gt; 0, ... JEE Main 2020 (Online) 4th September Morning Slot If y2 + loge (cos2x) = y,$$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$, then : ... JEE Main 2020 (Online) 3rd September Morning Slot Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b$$ \in $$R, g'(a)... JEE Main 2020 (Online) 9th January Evening Slot If$$x = 2\sin \theta - \sin 2\theta $$and$$y = 2\cos \theta - \cos 2\theta $$,$$\theta \in \left[ {0,2\pi } \righ...
JEE Main 2020 (Online) 9th January Evening Slot
If c is a point at which Rolle's theorem holds for the function, f(x) = $${\log _e}\left( {{{{x^2} + \alpha } \over {7x}... JEE Main 2020 (Online) 8th January Morning Slot Let y = y(x) be a function of x satisfying$$y\sqrt {1 - {x^2}} = k - x\sqrt {1 - {y^2}} $$where k is a constant and ... JEE Main 2020 (Online) 7th January Evening Slot If$$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) +...
JEE Main 2020 (Online) 7th January Morning Slot
The derivative of $${\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)$$, with respect to $${x \... JEE Main 2019 (Online) 12th April Evening Slot If ey + xy = e, the ordered pair$$\left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right)$$at x = 0 is equal to ... JEE Main 2019 (Online) 12th April Morning Slot If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is : JEE Main 2019 (Online) 8th April Evening Slot If$$2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)...
JEE Main 2019 (Online) 8th April Morning Slot
For x &gt; 1, if (2x)2y = 4e2x$$-$$2y, then (1 + loge 2x)2 $${{dy} \over {dx}}$$ is equal to : ...
JEE Main 2019 (Online) 12th January Morning Slot
If&nbsp;&nbsp;&nbsp;x $$=$$ 3 tan t and y $$=$$ 3 sec t, then the value of $${{{d^2}y} \over {d{x^2}}}$$ at t $$= {\pi... JEE Main 2019 (Online) 9th January Evening Slot If$$x=-1$$and$$x=2$$are extreme points of$$f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x$$the... JEE Main 2014 (Offline) If$$g$$is the inverse of a function$$f$$and$$f'\left( x \right) = {1 \over {1 + {x^5}}},$$then$$g'\left( x \right...
JEE Main 2014 (Offline)
If $$y = \sec \left( {{{\tan }^{ - 1}}x} \right),$$ then $${{{dy} \over {dx}}}$$ at $$x=1$$ is equal to :
JEE Main 2013 (Offline)
$${{{d^2}x} \over {d{y^2}}}$$ equals:
AIEEE 2011
Let $$f:\left( { - 1,1} \right) \to R$$ be a differentiable function with $$f\left( 0 \right) = - 1$$ and $$f'\left( 0... AIEEE 2010 Let$$y$$be an implicit function of$$x$$defined by$${x^{2x}} - 2{x^x}\cot \,y - 1 = 0$$. Then$$y'(1)$$equals AIEEE 2009 If$${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$then$${{{dy} \over {dx}}}$$is AIEEE 2006 The set of points where$$f\left( x \right) = {x \over {1 + \left| x \right|}}$$is differentiable is AIEEE 2006 The value of$$a$$for which the sum of the squares of the roots of the equation$${x^2} - \left( {a - 2} \right)x - a ...
AIEEE 2005
If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals
AIEEE 2005
Let $$f:R \to R$$ be a differentiable function having $$f\left( 2 \right) = 6$$, $$f'\left( 2 \right) = \left( {{1 \o... AIEEE 2005 If$$x = {e^{y + {e^y} + {e^{y + .....\infty }}}}$$,$$x &gt; 0,$$then$${{{dy} \over {dx}}}$$is AIEEE 2004 Let$$f\left( x \right)$$be a polynomial function of second degree. If$$f\left( 1 \right) = f\left( { - 1} \right)$$... AIEEE 2003 If$$f\left( x \right) = {x^n},$$then the value of$$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\lef...
AIEEE 2003
If $$f\left( y \right) = {e^y},$$ $$g\left( y \right) = y;y &gt; 0$$ and $$F\left( t \right) = \int\limits_0^t {f\left(... AIEEE 2003 If$$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},$$then$$\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{...
AIEEE 2002

## Numerical

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Let $$f:[0,1] \rightarrow \mathbf{R}$$ be a twice differentiable function in $$(0,1)$$ such that $$f(0)=3$$ and $$f(1)=5... JEE Main 2022 (Online) 28th July Morning Shift Let$$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8...
JEE Main 2022 (Online) 25th July Morning 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) 27th June Evening Shift
Let f and g be twice differentiable even functions on ($$-$$2, 2) such that $$f\left( {{1 \over 4}} \right) = 0$$, $$f\l... JEE Main 2022 (Online) 29th June Evening Shift Let f(x) be a cubic polynomial with f(1) =$$-$$10, f($$-$$1) = 6, and has a local minima at x = 1, and f'(x) has a loca... JEE Main 2021 (Online) 31st August Evening Shift If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greate... JEE Main 2021 (Online) 31st August Morning Shift If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then$${{{d^2}y} \over {d{x^2}}}$$at x = 0 is equ... JEE Main 2021 (Online) 26th August Morning Shift If$$f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)$$and its first... JEE Main 2021 (Online) 17th March Morning Shift If y =$$\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}} $$, then$$...
JEE Main 2020 (Online) 2nd September Evening Slot

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