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Differentiation

81 questions · Mathematics · JEE Main

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JEE Main
Mathematics
Differentiation

Differentiation

81 questions · Mathematics · JEE Main

Let $f : R \to R$ be a twice differentiable function such that $(sin x cos y) (f (2 x + 2 y) - f (2 x - 2 y)) = (cos x sin y) (f (2 x + 2 y) + f (2 x - 2 y))$ , for all $x, y \in R$ . If $f^{'} (0) = \frac{1}{2}$ , then…2025 · 2 Apr · Shift 1 · Q40 · MCQ
Let $f : R \to R$ be a thrice differentiable odd function satisfying $f^{'} (x) \geq 0, f^{'} (x) = f (x), f (0) = 0, f^{'} (0) = 3$ . Then $9 f (lo g_{e} 3)$ is equal to .2025 · 2 Apr · Shift 1 · Q47 · Numerical
$If y (x) = sin x 271 cos x 281 sin x + cos x + 1 271, x \in R, then \frac{d ^{2} y}{d x ^{2}} + y is equal to$ 2025 · 3 Apr · Shift 1 · Q43 · MCQ
Let $f : (0, \infty) \to R$ be a function which is differentiable at all points of its domain and satisfies the condition $x^{2} f^{'} (x) = 2 x f (x) + 3$ , with $f (1) = 4$ . Then $2 f (2)$ is equal to :2025 · 24 Jan · Shift 2 · Q44 · MCQ
If $y = \frac{( x + 1 ) ( x ^{2} - x )}{x x + x + x} + \frac{1}{15} (3 cos^{2} x - 5) cos^{3} x$ , then $96 y^{'} (\frac{π}{6})$ is equal to :2024 · 1 Feb · Shift 2 · Q53 · Numerical
Let $f (x) = x^{5} + 2 e^{x /4}$ for all $x \in R$ . Consider a function $g (x)$ such that $(g \circ f) (x) = x$ for all $x \in R$ . Then the value of $8 g^{'} (2)$ is :2024 · 4 Apr · Shift 1 · Q37 · MCQ
Let $f : R \to R$ be a thrice differentiable function such that $f (0) = 0, f (1) = 1, f (2) = - 1, f (3) = 2$ and $f (4) = - 2$ . Then, the minimum number of zeros of $(3 f^{'} f^{''} + f f^{'''}) (x)$ …2024 · 4 Apr · Shift 2 · Q58 · Numerical
If $y (θ) = \frac{2 c o s θ + c o s 2 θ}{c o s 3 θ + 4 c o s 2 θ + 5 c o s θ + 2}$ , then at $θ = \frac{π}{2}, y^{''} + y^{'} + y$ is equal to :2024 · 5 Apr · Shift 2 · Q40 · MCQ
$If f (x) = {x^{3} sin (\frac{1}{x}), 0 x e q 0, x = 0, then$ 2024 · 6 Apr · Shift 1 · Q38 · MCQ
Let $f : (- \infty, \infty) - {0} \to R$ be a differentiable function such that $f^{'} (1) = lim_{a \to \infty} a^{2} f (\frac{1}{a})$ . Then $lim_{a \to \infty} \frac{a ( a + 1 )}{2} tan^{- 1} (\frac{1}{a}) + a^{2} - 2 lo g_{e} a$ …2024 · 6 Apr · Shift 1 · Q43 · MCQ
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:2024 · 6 Apr · Shift 2 · Q46 · MCQ
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:2024 · 9 Apr · Shift 1 · Q33 · MCQ
If $lo g_{e} y = 3 sin^{- 1} x$ , then $(1 - x^{2}) y^{''} - x y^{'}$ at $x = \frac{1}{2}$ is equal to2024 · 9 Apr · Shift 2 · Q38 · MCQ
Let $f (x) = x^{3} + x^{2} f^{'} (1) + x f^{''} (2) + f^{'''} (3), x \in R$ . Then $f^{'} (10)$ is equal to .2024 · 27 Jan · Shift 1 · Q53 · Numerical
Suppose $f (x) = \frac{( 2 ^{x} + 2 ^{- x} ) t a n x t a n ^{- 1} ( x ^{2} - x + 1 )}{( 7 x ^{2} + 3 x + 1 ) ^{3}}$ . Then the value of $f^{'} (0)$ is equal to2024 · 29 Jan · Shift 1 · Q31 · MCQ
$Let y = lo g_{e} (\frac{1 - x ^{2}}{1 + x ^{2}}), - 1 < x < 1 . Then at x = \frac{1}{2}, the value of 225 (y^{'} - y^{''}) is equal to$ 2024 · 29 Jan · Shift 2 · Q40 · MCQ
Let $g : R \to R$ be a non constant twice differentiable function such that $g^{'} (\frac{1}{2}) = g^{'} (\frac{3}{2})$ . If a real valued function $f$ is defined as…2024 · 30 Jan · Shift 1 · Q31 · MCQ
If $f (x) = 2 cos^{4} x 3 + 2 cos^{4} x 2 cos^{4} x 2 sin^{4} x 2 sin^{4} x 3 + 2 sin^{4} x 3 + sin^{2} 2 x sin^{2} 2 x sin^{2} 2 x,$ then $\frac{1}{5} f^{'} (0) =$ …2024 · 30 Jan · Shift 1 · Q36 · MCQ
Let $f : R - {0} \to R$ be a function satisfying $f (\frac{x}{y}) = \frac{f ( x )}{f ( y )}$ for all $x, y, f (y) e q 0$ . If $f^{'} (1) = 2024$ , then2024 · 30 Jan · Shift 2 · Q36 · MCQ
Let $f (x) = 2 x + tan^{- 1} x$ and $g (x) = lo g_{e} (1 + x^{2} + x), x \in [0, 3]$ . Then2023 · 1 Feb · Shift 1 · Q30 · MCQ
If $f (x) = x^{2} + g^{'} (1) x + g^{''} (2)$ and $g (x) = f (1) x^{2} + x f^{'} (x) + f^{''} (x)$ , then the value of $f (4) - g (4)$ is equal to .2023 · 1 Feb · Shift 1 · Q43 · Numerical
If $y (x) = x^{x}, x > 0$ , then $y^{''} (2) - 2 y^{'} (2)$ is equal to2023 · 1 Feb · Shift 2 · Q29 · MCQ
If $2 x^{y} + 3 y^{x} = 20$ , then $\frac{d y}{d x}$ at $(2, 2)$ is equal to :2023 · 6 Apr · Shift 1 · Q35 · MCQ
Let $f (x) = \frac{s i n x + c o s x - 2}{s i n x - c o s x}, x \in [0, π] - {\frac{π}{4}}$ . Then $f (\frac{7 π}{12}) f^{''} (\frac{7 π}{12})$ is equal to2023 · 8 Apr · Shift 1 · Q23 · MCQ
For the differentiable function $f : R - {0} \to R$ , let $3 f (x) + 2 f (\frac{1}{x}) = \frac{1}{x} - 10$ , then $f (3) + f^{'} (\frac{1}{4})$ is equal to2023 · 13 Apr · Shift 1 · Q35 · MCQ
Let $f (x) = \sum_{k = 1}^{10} k x^{k}, x \in R$ . If $2 f (2) + f^{'} (2) = 119 (2)^{n} + 1$ then $n$ is equal to 2023 · 13 Apr · Shift 2 · Q39 · Numerical
If $f (x) = x^{3} - x^{2} f^{'} (1) + x f^{''} (2) - f^{'''} (3), x \in R$ , then2023 · 24 Jan · Shift 2 · Q35 · MCQ
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 to2023 · 25 Jan · Shift 1 · Q32 · MCQ
Let $f : R \to R$ be a differentiable function that satisfies the relation $f (x + y) = f (x) + f (y) - 1, \forall x, y \in R$ . If $f^{'} (0) = 2$ , then $∣ f (- 2) ∣$ is equal to .2023 · 29 Jan · Shift 1 · Q46 · Numerical
Let $f$ and $g$ be the twice differentiable functions on $R$ such that $f^{''} (x) = g^{''} (x) + 6 x f^{'} (1) = 4 g^{'} (1) - 3 = 9 f (2) = 3 g (2) = 12$ . Then which of the following is NOT true?2023 · 29 Jan · Shift 2 · Q38 · MCQ
Let $f^{1} (x) = \frac{3 x + 2}{2 x + 3}, x \in R - {\frac{- 3}{2}}$ For $n \geq 2$ , define $f^{n} (x) = f^{1} o f^{n - 1} (x)$ . If $f^{5} (x) = \frac{a x + b}{b x + a}, gcd (a, b) = 1$ …2023 · 30 Jan · Shift 1 · Q38 · Numerical
Let $y = f (x) = sin^{3} (\frac{π}{3} (cos (\frac{π}{3 2} (- 4 x^{3} + 5 x^{2} + 1)^{\frac{3}{2}})))$ . Then, at x = 1,2023 · 31 Jan · Shift 1 · Q39 · MCQ
If $y = tan^{- 1} (sec x^{3} - tan x^{3}), \frac{π}{2} < x^{3} < \frac{3 π}{2}$ , then2022 · 24 Jun · Shift 2 · Q35 · MCQ
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 .2022 · 25 Jun · Shift 1 · Q27 · MCQ
The value of $lo g_{e} 2 \frac{d}{d x} (lo g_{c o s x} cosec x)$ at $x = \frac{π}{4}$ is2022 · 26 Jul · Shift 2 · Q26 · MCQ
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 $14. \frac{f ^{'} ( 4 )}{f ^{'} ( 2 )}$ is equal to .2022 · 26 Jun · Shift 2 · Q39 · Numerical
For the curve $C : (x^{2} + y^{2} - 3) + (x^{2} - y^{2} - 1)^{5} = 0$ , the value of $3 y^{'} - y^{3} y^{''}$ , at the point $(α, α)$ , $α > 0$ , on C, is equal to .2022 · 27 Jul · Shift 2 · Q37 · Numerical
If $cos^{- 1} (\frac{y}{2}) = lo g_{e} (\frac{x}{5})^{5}, ∣ y ∣ < 2$ , then :2022 · 27 Jun · Shift 1 · Q25 · MCQ
If $y (x) = (x^{x})^{x}, x > 0$ , then $\frac{d ^{2} x}{d y ^{2}} + 20$ at x = 1 is equal to .2022 · 27 Jun · Shift 2 · Q41 · Numerical
Let $x (t) = 22 cos t sin 2 t$ and $y (t) = 22 sin t sin 2 t, t \in (0, \frac{π}{2})$ . Then $\frac{1 + ( \frac{d y}{d x} ) ^{2}}{\frac{d ^{2} y}{d x ^{2}}}$ at $t = \frac{π}{4}$ is equal to :2022 · 28 Jul · Shift 2 · Q28 · MCQ
Let f and g be twice differentiable even functions on ( $-$ 2, 2) such that $f (\frac{1}{4}) = 0$ , $f (\frac{1}{2}) = 0$ , $f (1) = 1$ and $g (\frac{3}{4}) = 0$ , $g (1) = 2$ . Then, the minimum…2022 · 29 Jun · Shift 2 · Q39 · Numerical
If $f (x) = sin (cos^{- 1} (\frac{1 - 2 ^{2 x}}{1 + 2 ^{2 x}}))$ and its first derivative with respect to x is $- \frac{b}{a} lo g_{e} 2$ when x = 1, where a and b are integers, then the minimum…2021 · 17 Mar · Shift 1 · Q41 · Numerical
Let $f (x) = cos (2 tan^{- 1} sin (cot^{- 1} \frac{1 - x}{x}))$ , 0 < x < 1. Then :2021 · 26 Aug · Shift 1 · Q27 · MCQ
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then $\frac{d ^{2} y}{d x ^{2}}$ at x = 0 is equal to .2021 · 26 Aug · Shift 1 · Q40 · Numerical
If $y (x) = cot^{- 1} (\frac{1 + s i n x + 1 - s i n x}{1 + s i n x - 1 - s i n x}), x \in (\frac{π}{2}, π)$ , then $\frac{d y}{d x}$ at $x = \frac{5 π}{6}$ is :2021 · 27 Aug · Shift 2 · Q33 · MCQ
If y = $k = 1 \sum 6 k cos^{- 1} {\frac{3}{5} cos k x - \frac{4}{5} sin k x}$ , then $\frac{d y}{d x}$ at x = 0 is .2020 · 2 Sep · Shift 2 · Q33 · Numerical
If y2 + loge (cos2x) = y, $x \in (- \frac{π}{2}, \frac{π}{2})$ , then :2020 · 3 Sep · Shift 1 · Q28 · MCQ
If $(a + 2 b cos x) (a - 2 b cos y) = a^{2} - b^{2}$ where a > b > 0, then $\frac{d x}{d y} a t (\frac{π}{4}, \frac{π}{4})$ is :2020 · 4 Sep · Shift 1 · Q27 · MCQ
The derivative of $tan^{- 1} (\frac{1 + x ^{2} - 1}{x})$ with respect to $tan^{- 1} (\frac{2 x 1 - x ^{2}}{1 - 2 x ^{2}})$ at x = $\frac{1}{2}$ is :2020 · 5 Sep · Shift 2 · Q33 · MCQ
Let xk + yk = ak, (a, k > 0 ) and $\frac{d y}{d x} + (\frac{y}{x})^{\frac{1}{3}} = 0$ , then k is:2020 · 7 Jan · Shift 1 · Q28 · MCQ
If $y (α) = 2 (\frac{t a n α + c o t α}{1 + t a n ^{2} α}) + \frac{1}{s i n ^{2} α}, α \in (\frac{3 π}{4}, π) \frac{d y}{d α} a t α = \frac{5 π}{6} i s$ …2020 · 7 Jan · Shift 1 · Q36 · MCQ
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 $y (\frac{1}{2}) = - \frac{1}{4}$ . Then $\frac{d y}{d x}$ at x = $\frac{1}{2}$ , is equal to :2020 · 7 Jan · Shift 2 · Q34 · MCQ
Let ƒ(x) = (sin(tan–1x) + sin(cot–1x))2 – 1, |x| > 1. If $\frac{d y}{d x} = \frac{1}{2} \frac{d}{d x} (sin^{- 1} (f (x)))$ and $y (3) = \frac{π}{6}$ , then y( $- 3$ …2020 · 8 Jan · Shift 1 · Q36 · MCQ
If $x = 2 sin θ - sin 2 θ$ and $y = 2 cos θ - cos 2 θ$ , $θ \in [0, 2 π]$ , then $\frac{d ^{2} y}{d x ^{2}}$ at $θ$ = $π$ is :2020 · 9 Jan · Shift 2 · Q33 · MCQ
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) is equal to :2020 · 9 Jan · Shift 2 · Q39 · MCQ
If $2 y = (cot^{- 1} (\frac{3 c o s x + s i n x}{c o s x - 3 s i n x}))^{2}$ , x $\in$ $(0, \frac{π}{2})$ then $\frac{d y}{d x}$ is equal to:2019 · 8 Apr · Shift 1 · Q43 · MCQ
If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is :2019 · 8 Apr · Shift 2 · Q35 · MCQ
If x $=$ 3 tan t and y $=$ 3 sec t, then the value of $\frac{d ^{2} y}{d x ^{2}}$ at t $= \frac{π}{4},$ is :2019 · 9 Jan · Shift 2 · Q30 · MCQ
Let f(x) = loge(sin x), (0 < x < $π$ ) and g(x) = sin–1 (e–x ), (x $\geq$ 0). If $α$ is a positive real number such that a = (fog)'( $α$ ) and b = (fog)( $α$ ), then :2019 · 10 Apr · Shift 2 · Q23 · MCQ
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 -2019 · 10 Jan · Shift 1 · Q43 · MCQ
If xloge(logex) $-$ x2 + y2 = 4(y > 0), then $\frac{d y}{d x}$ at x = e is equal to :2019 · 11 Jan · Shift 1 · Q21 · MCQ
If ey + xy = e, the ordered pair $(\frac{d y}{d x}, \frac{d ^{2} y}{d x ^{2}})$ at x = 0 is equal to :2019 · 12 Apr · Shift 1 · Q23 · MCQ
The derivative of $tan^{- 1} (\frac{s i n x - c o s x}{s i n x + c o s x})$ , with respect to $\frac{x}{2}$ , where $(x \in (0, \frac{π}{2}))$ is :2019 · 12 Apr · Shift 2 · Q34 · MCQ
For x > 1, if (2x)2y = 4e2x $-$ 2y, then (1 + loge 2x)2 $\frac{d y}{d x}$ is equal to :2019 · 12 Jan · Shift 1 · Q41 · MCQ
If x2 + y2 + sin y = 4, then the value of $\frac{d ^{2} y}{d x ^{2}}$ at the point ( $-$ 2,0) is :2018 · 15 Apr · Shift 1 · Q46 · MCQ
If $f (x) = cos x 2 sin x tan x x x^{2} x 1 2 x 1,$ then $x \to 0 lim \frac{f ^{'} ( x )}{x}$ 2018 · 15 Apr · Shift 1 · Q47 · MCQ
If f(x) = sin-1 $(\frac{2 \times 3 ^{x}}{1 + 9 ^{x}}),$ then f' $(- \frac{1}{2})$ equals :2018 · 15 Apr · Shift 2 · Q42 · MCQ
If $x = 2^{c o s e c^{- 1}}$ and $y = 2^{se c^{- 1} t} (∣ t ∣ \geq 1),$ then $\frac{d y}{d x}$ is equal to :2018 · 16 Apr · Shift 1 · Q28 · MCQ
If y = $[x + x^{2} - 1]^{15} + [x - x^{2} - 1]^{15},$ then (x2 $-$ 1) $\frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x}$ is equal to :2017 · 8 Apr · Shift 1 · Q47 · MCQ
Let f be a polynomial function such that f (3x) = f ' (x) . f '' (x), for all x $\in$ R. Then :2017 · 9 Apr · Shift 1 · Q42 · MCQ
If for $x \in (0, \frac{1}{4})$ , the derivatives of $tan^{- 1} (\frac{6 x x}{1 - 9 x ^{3}})$ is $x . g (x)$ , then $g (x)$ equals2017 · Shift 0 · Q42 · MCQ
If $g$ is the inverse of a function $f$ and $f^{'} (x) = \frac{1}{1 + x ^{5}},$ then $g^{'} (x)$ is equal to:2014 · Shift 0 · Q34 · MCQ
If $y = sec (tan^{- 1} x),$ then $\frac{d y}{d x}$ at $x = 1$ is equal to :2013 · Shift 0 · Q35 · MCQ
$\frac{d ^{2} x}{d y ^{2}}$ equals:2011 · Shift 0 · Q43 · MCQ
Let $f : (- 1, 1) \to R$ be a differentiable function with $f (0) = - 1$ and $f^{'} (0) = 1$ . Let $g (x) = [f (2 f (x) + 2)]^{2}$ . Then $g^{'} (0) =$ …2010 · Shift 0 · Q40 · MCQ
Let $y$ be an implicit function of $x$ defined by $x^{2 x} - 2 x^{x} cot y - 1 = 0$ . Then $y^{'} (1)$ equals2009 · Shift 0 · Q31 · MCQ
If $x^{m} . y^{n} = (x + y)^{m + n},$ then $\frac{d y}{d x}$ is2006 · Shift 0 · Q56 · MCQ
If $x = e^{y + e^{y} + e^{y + .....\infty}}$ , $x > 0,$ then $\frac{d y}{d x}$ is2004 · Shift 0 · Q77 · MCQ
If $f (x) = x^{n},$ then the value of $f (1) - \frac{f ^{'} ( 1 )}{1 !} + \frac{f ^{''} ( 1 )}{2 !} - \frac{f ^{'''} ( 1 )}{3 !} + .......... \frac{( - 1 ) ^{n} f ^{n} ( 1 )}{n !}$ …2003 · Shift 0 · Q116 · MCQ
Let $f (x)$ be a polynomial function of second degree. If $f (1) = f (- 1)$ and $a, b, c$ are in $A . P,$ then $f^{'} (a), f^{'} (b), f^{'} (c)$ are in2003 · Shift 0 · Q117 · MCQ
If $y = (x + 1 + x^{2})^{n},$ then $(1 + x^{2}) \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x}$ is2002 · Shift 0 · Q75 · MCQ