Question 1: Palette 1/68

JEE Mains 2026 MCQ

Let $f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)$, $x \in \mathbb{R}$. Then the value of $f'(5)$ is :

A

$\dfrac{657}{5}$

B

$\dfrac{117}{5}$

C

$\dfrac{2}{5}$

D

$\dfrac{62}{5}$

Question 2: Palette 2/68

JEE Mains 2026 MCQ

If $y=\tan ^{-1}\left(\frac{3 \cos x-4 \sin x}{4 \cos x+3 \sin x}\right)+2 \tan ^{-1}\left(\frac{x}{1+\sqrt{1-x^2}}\right)$, then $\frac{d y}{d x}$ at $x=\frac{\sqrt{3}}{2}$ is equal to :

A

3

B

-1

C

1

D

2

Question 3: Palette 3/68

JEE Mains 2025 MCQ

$ \text { If } y(x)=\left|\begin{array}{ccc} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{array}\right|, x \in \mathbb{R} \text {, then } \frac{d^2 y}{d x^2}+y \text { is equal to } $

A

28

B

27

C

-1

D

1

Question 4: Palette 4/68

JEE Mains 2025 MCQ

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ 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 $x, y \in \mathbf{R}$. If $f^{\prime}(0)=\frac{1}{2}$, then the value of $24 f^{\prime \prime}\left(\frac{5 \pi}{3}\right)$ is :

A

2

B

3

C

$-$3

D

$-$2

Question 5: Palette 5/68

JEE Mains 2025 MCQ

Let $f:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f^{\prime}(x)=2 x f(x)+3$, with $f(1)=4$. Then $2 f(2)$ is equal to :

A

19

B

23

C

29

D

39

Question 6: Palette 6/68

JEE Mains 2024 MCQ

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

A

$9 e^{\pi / 2}$

B

$9 e^{\pi / 6}$

C

$3 e^{\pi / 2}$

D

$3 e^{\pi / 6}$

Question 7: Palette 7/68

JEE Mains 2024 MCQ

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:

A

54

B

51

C

73

D

62

Question 8: Palette 8/68

JEE Mains 2024 MCQ

Suppose for a differentiable function $h, h(0)=0, h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $g(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$, then $g^{\prime}(0)$ is equal to:

A

4

B

5

C

3

D

8

Question 9: Palette 9/68

JEE Mains 2024 MCQ

$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\right. \text {, then }$

A

$f^{\prime \prime}(0)=0$

B

$f^{\prime \prime}(0)=1$

C

$f^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{24-\pi^2}{2 \pi}$

D

$f^{\prime \prime}\left(\frac{2}{\pi}\right)=\frac{12-\pi^2}{2 \pi}$

Question 10: Palette 10/68

JEE Mains 2024 MCQ

Let $f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _\limits{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim _\limits{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a$ is equal to

A

$\frac{5}{2}+\frac{\pi}{8}$

B

$\frac{3}{8}+\frac{\pi}{4}$

C

$\frac{3}{4}+\frac{\pi}{8}$

D

$\frac{3}{2}+\frac{\pi}{4}$

Question 11: Palette 11/68

JEE Mains 2024 MCQ

If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$, then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal to :

A

$\frac{1}{2}$

B

1

C

$\frac{3}{2}$

D

2

Question 12: Palette 12/68

JEE Mains 2024 MCQ

Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

A

4

B

2

C

16

D

8

Question 13: Palette 13/68

JEE Mains 2024 MCQ

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 $x, y, f(y) \neq 0$. If $f^{\prime}(1)=2024$, then

A

$x f^{\prime}(x)+2024 f(x)=0$

B

$x f^{\prime}(x)-2023 f(x)=0$

C

$x f^{\prime}(x)-2024 f(x)=0$

D

$x f^{\prime}(x)+f(x)=2024$

Question 14: Palette 14/68

JEE Mains 2024 MCQ

Let $g: \mathbf{R} \rightarrow \mathbf{R}$ be a non constant twice differentiable function such that $\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then

A

$f^{\prime \prime}(x)=0$ for atleast two $x$ in $(0,2)$

B

$f^{\prime}\left(\frac{3}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)=1$

C

$f^{\prime \prime}(x)=0$ for no $x$ in $(0,1)$

D

$f^{\prime \prime}(x)=0$ for exactly one $x$ in $(0,1)$

Question 15: Palette 15/68

JEE Mains 2024 MCQ

If $f(x)=\left|\begin{array}{ccc} 2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x \end{array}\right|,$ then $\frac{1}{5} f^{\prime}(0)=$ is equal to :

A

2

B

1

C

0

D

6

Question 16: Palette 16/68

JEE Mains 2024 MCQ

$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 < x<1 \text {. Then at } x=\frac{1}{2} \text {, the value of } 225\left(y^{\prime}-y^{\prime \prime}\right) \text { is equal to }$

A

732

B

736

C

742

D

746

Question 17: Palette 17/68

JEE Mains 2024 MCQ

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 $f^{\prime}(0)$ is equal to

A

$\pi$

B

$\sqrt{\pi}$

C

0

D

$\frac{\pi}{2}$

Question 18: Palette 18/68

JEE Mains 2023 MCQ

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 $\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|$ is equal to

A

13

B

$\frac{29}{5}$

C

$\frac{33}{5}$

D

7

Question 19: Palette 19/68

JEE Mains 2023 MCQ

Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)$ is equal to

A

$\frac{2}{3 \sqrt{3}}$

B

$\frac{2}{9}$

C

$\frac{-1}{3 \sqrt{3}}$

D

$\frac{-2}{3}$

Question 20: Palette 20/68

JEE Mains 2023 MCQ

If $2 x^{y}+3 y^{x}=20$, then $\frac{d y}{d x}$ at $(2,2)$ is equal to :

A

$-\left(\frac{3+\log _{e} 16}{4+\log _{e} 8}\right)$

B

$-\left(\frac{2+\log _{e} 8}{3+\log _{e} 4}\right)$

C

$-\left(\frac{3+\log _{e} 8}{2+\log _{e} 4}\right)$

D

$-\left(\frac{3+\log _{e} 4}{2+\log _{e} 8}\right)$

Question 21: Palette 21/68

JEE Mains 2023 MCQ

If $y(x)=x^{x},x > 0$, then $y''(2)-2y'(2)$ is equal to

A

$4(\log_{e}2)^{2}+2$

B

$8\log_{e}2-2$

C

$4\log_{e}2+2$

D

$4(\log_{e}2)^{2}-2$

Question 22: Palette 22/68

JEE Mains 2023 MCQ

Let $f(x) = 2x + {\tan ^{ - 1}}x$ and $g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$. Then

A

there exists $\widehat x \in [0,3]$ such that $f'(\widehat x) < g'(\widehat x)$

B

there exist $0 < {x_1} < {x_2} < 3$ such that $f(x) < g(x),\forall x \in ({x_1},{x_2})$

C

$\min f'(x) = 1 + \max g'(x)$

D

$\max f(x) > \max g(x)$

Question 23: Palette 23/68

JEE Mains 2023 MCQ

Let $y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$. Then, at x = 1,

A

$2 y^{\prime}+\sqrt{3} \pi^{2} y=0$

B

$y^{\prime}+3 \pi^{2} y=0$

C

$\sqrt{2} y^{\prime}-3 \pi^{2} y=0$

D

$2 y^{\prime}+3 \pi^{2} y=0$

Question 24: Palette 24/68

JEE Mains 2023 MCQ

Let $f$ and $g$ be the twice differentiable functions on $\mathbb{R}$ such that

$f''(x)=g''(x)+6x$

$f'(1)=4g'(1)-3=9$

$f(2)=3g(2)=12$.

Then which of the following is NOT true?

A

$g(-2)-f(-2)=20$

B

There exists $x_0\in(1,3/2)$ such that $f(x_0)=g(x_0)$

C

$|f'(x)-g'(x)| < 6\Rightarrow -1 < x < 1$

D

If $-1 < x < 2$, then $|f(x)-g(x)| < 8$

Question 25: Palette 25/68

JEE Mains 2023 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 to

A

496

B

976

C

464

D

944

Question 26: Palette 26/68

JEE Mains 2023 MCQ

If $f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$, then

A

$2f(0) - f(1) + f(3) = f(2)$

B

$f(1) + f(2) + f(3) = f(0)$

C

$f(3) - f(2) = f(1)$

D

$3f(1) + f(2) = f(3)$

Question 27: Palette 27/68

JEE Mains 2022 MCQ

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 $\frac{1+\left(\frac{d y}{d x}\right)^{2}}{\frac{d^{2} y}{d x^{2}}}$ at $t=\frac{\pi}{4}$ is equal to :

A

$\frac{-2 \sqrt{2}}{3}$

B

$\frac{2}{3}$

C

$\frac{1}{3}$

D

$ \frac{-2}{3}$

Question 28: Palette 28/68

JEE Mains 2022 MCQ

The value of $\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$ at $x=\frac{\pi}{4}$ is

A

$-2 \sqrt{2}$

B

$2 \sqrt{2}$

C

$-4$

D

4

Question 29: Palette 29/68

JEE Mains 2022 MCQ

If ${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y| < 2$, then :

A

${x^2}y'' + xy' - 25y = 0$

B

${x^2}y'' - xy' - 25y = 0$

C

${x^2}y'' - xy' + 25y = 0$

D

${x^2}y'' + xy' + 25y = 0$

Question 30: Palette 30/68

JEE Mains 2022 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 ________________.

A

${1 \over {49}}$

B

${3 \over {49}}$

C

${43 \over {49}}$

D

${91 \over {49}}$

Question 31: Palette 31/68

JEE Mains 2022 MCQ

If $y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2} < {x^3} < {{3\pi } \over 2}$, then

A

$xy'' + 2y' = 0$

B

${x^2}y'' - 6y + {{3\pi } \over 2} = 0$

C

${x^2}y'' - 6y + 3\pi = 0$

D

$xy'' - 4y' = 0$

Question 32: Palette 32/68

JEE Mains 2021 MCQ

If $y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left( {{\pi \over 2},\pi } \right)$, then ${{dy} \over {dx}}$ at $x = {{5\pi } \over 6}$ is :

A

$ - {1 \over 2}$

B

$-$1

C

${1 \over 2}$

D

0

Question 33: Palette 33/68

JEE Mains 2021 MCQ

Let $f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$, 0 < x < 1. Then :

A

${(1 - x)^2}f'(x) - 2{(f(x))^2} = 0$

B

${(1 + x)^2}f'(x) + 2{(f(x))^2} = 0$

C

${(1 - x)^2}f'(x) + 2{(f(x))^2} = 0$

D

${(1 + x)^2}f'(x) - 2{(f(x))^2} = 0$

Question 34: Palette 34/68

JEE Mains 2020 MCQ

The derivative of
${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$ with
respect to ${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}} } \over {1 - 2{x^2}}}} \right)$ at x = ${1 \over 2}$ is :

A

${{2\sqrt 3 } \over 3}$

B

${{2\sqrt 3 } \over 5}$

C

${{\sqrt 3 } \over {10}}$

D

${{\sqrt 3 } \over {12}}$

Question 35: Palette 35/68

JEE Mains 2020 MCQ

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 ${{dx} \over {dy}}\,\,at\left( {{\pi \over 4},{\pi \over 4}} \right)$ is :

A

${{a - 2b} \over {a + 2b}}$

B

${{a - b} \over {a + b}}$

C

${{a + b} \over {a - b}}$

D

${{2a + b} \over {2a - b}}$

Question 36: Palette 36/68

JEE Mains 2020 MCQ

If y² + log_e (cos²x) = y,
$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$, then :

A

|y''(0)| = 2

B

|y'(0)| + |y''(0)| = 3

C

y''(0) = 0

D

|y'(0)| + |y"(0)| = 1

Question 37: Palette 37/68

JEE Mains 2020 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 :

A

1

B

5

C

${2 \over 5}$

D

${1 \over 5}$

Question 38: Palette 38/68

JEE Mains 2020 MCQ

If $x = 2\sin \theta - \sin 2\theta $ and $y = 2\cos \theta - \cos 2\theta $,
$\theta \in \left[ {0,2\pi } \right]$, then ${{{d^2}y} \over {d{x^2}}}$ at $\theta $ = $\pi $ is :

A

${3 \over 8}$

B

${3 \over 2}$

C

${3 \over 4}$

D

-${3 \over 4}$

Question 39: Palette 39/68

JEE Mains 2020 MCQ

Let ƒ(x) = (sin(tan^–1x) + sin(cot^–1x))² – 1, |x| > 1.
If ${{dy} \over {dx}} = {1 \over 2}{d \over {dx}}\left( {{{\sin }^{ - 1}}\left( {f\left( x \right)} \right)} \right)$ and $y\left( {\sqrt 3 } \right) = {\pi \over 6}$, then y(${ - \sqrt 3 }$) is equal to :

A

${{5\pi } \over 6}$

B

$ - {\pi \over 6}$

C

${\pi \over 3}$

D

${{2\pi } \over 3}$

Question 40: Palette 40/68

JEE Mains 2020 MCQ

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

$y\left( {{1 \over 2}} \right) = - {1 \over 4}$. Then ${{dy} \over {dx}}$ at x = ${1 \over 2}$, is equal to :

A

${2 \over {\sqrt 5 }}$

B

$ - {{\sqrt 5 } \over 2}$

C

${{\sqrt 5 } \over 2}$

D

$ - {{\sqrt 5 } \over 4}$

Question 41: Palette 41/68

JEE Mains 2020 MCQ

Let x^k + y^k = a^k, (a, k > 0 ) and ${{dy} \over {dx}} + {\left( {{y \over x}} \right)^{{1 \over 3}}} = 0$, then k is:

A

${1 \over 3}$

B

${2 \over 3}$

C

${4 \over 3}$

D

${3 \over 2}$

Question 42: Palette 42/68

JEE Mains 2020 MCQ

If $y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }}} ,\alpha \in \left( {{{3\pi } \over 4},\pi } \right)$

${{dy} \over {d\alpha }}\,\,at\,\alpha = {{5\pi } \over 6}is$ :

A

4

B

-4

C

${4 \over 3}$

D

-${1 \over 4}$

Question 43: Palette 43/68

JEE Mains 2019 MCQ

The derivative of ${\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)$, with respect to ${x \over 2}$ , where $\left( {x \in \left( {0,{\pi \over 2}} \right)} \right)$ is :

A

1

B

2

C

${2 \over 3}$

D

${1 \over 2}$

Question 44: Palette 44/68

JEE Mains 2019 MCQ

If e^y + xy = e, the ordered pair $\left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right)$ at x = 0 is equal to :

A

$\left( {{1 \over e}, - {1 \over {{e^2}}}} \right)$

B

$\left( { - {1 \over e},{1 \over {{e^2}}}} \right)$

C

$\left( { - {1 \over e}, - {1 \over {{e^2}}}} \right)$

D

$\left( {{1 \over e},{1 \over {{e^2}}}} \right)$

Question 45: Palette 45/68

JEE Mains 2019 MCQ

Let f(x) = log_e(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)'($\alpha $) and b = (fog)($\alpha $), then :

A

a$\alpha $² + b$\alpha $ - a = -2$\alpha $²

B

a$\alpha $² + b$\alpha $ + a = 0

C

a$\alpha $² - b$\alpha $ - a = 0

D

a$\alpha $² - b$\alpha $ - a = 1

Question 46: Palette 46/68

JEE Mains 2019 MCQ

If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))² at x = 1 is :

A

33

B

12

C

9

D

15

Question 47: Palette 47/68

JEE Mains 2019 MCQ

If $2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}$,

x $ \in $ $\left( {0,{\pi \over 2}} \right)$ then $dy \over dx$ is equal to:

A

$2x - {\pi \over 3}$

B

${\pi \over 6} - x$

C

${\pi \over 3} - x$

D

$x - {\pi \over 6}$

Question 48: Palette 48/68

JEE Mains 2019 MCQ

For x > 1, if (2x)^2y = 4e^2x$-$2y,

then (1 + log_e 2x)² ${{dy} \over {dx}}$ is equal to :

A

${{x\,{{\log }_e}2x - {{\log }_e}2} \over x}$

B

log_e 2x

C

x log_e 2x

D

${{x\,{{\log }_e}2x + {{\log }_e}2} \over x}$

Question 49: Palette 49/68

JEE Mains 2019 MCQ

If  xlog_e(log_ex) $-$ x² + y² = 4(y > 0), then ${{dy} \over {dx}}$ at x = e is equal to :

A

${{\left( {1 + 2e} \right)} \over {2\sqrt {4 + {e^2}} }}$

B

${{\left( {1 + 2e} \right)} \over {\sqrt {4 + {e^2}} }}$

C

${{\left( {2e - 1} \right)} \over {2\sqrt {4 + {e^2}} }}$

D

${e \over {\sqrt {4 + {e^2}} }}$

Question 50: Palette 50/68

JEE Mains 2019 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 -

A

30

B

$-$ 2

C

$-$ 4

D

8

Question 51: Palette 51/68

JEE Mains 2019 MCQ

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 :

A

${1 \over {3\sqrt 2 }}$

B

${1 \over {6\sqrt 2 }}$

C

${3 \over {2\sqrt 2 }}$

D

${1 \over 6}$

Question 52: Palette 52/68

JEE Mains 2018 MCQ

If $x = \sqrt {{2^{\cos e{c^{ - 1}}}}} $ and $y = \sqrt {{2^{se{c^{ - 1}}t}}} \,\,\left( {\left| t \right| \ge 1} \right),$ then ${{dy} \over {dx}}$ is equal to :

A

${y \over x}$

B

${x \over y}$

C

$-$ ${y \over x}$

D

$-$ ${x \over y}$

Question 53: Palette 53/68

JEE Mains 2018 MCQ

If    f(x) = sin^-1 $\left( {{{2 \times {3^x}} \over {1 + {9^x}}}} \right),$ then f'$\left( { - {1 \over 2}} \right)$ equals :

A

$ - \sqrt 3 {\log _e}\sqrt 3 $

B

$ \sqrt 3 {\log _e}\sqrt 3 $

C

$ - \sqrt 3 {\log _e}\, 3 $

D

$ \sqrt 3 {\log _e}\, 3 $

Question 54: Palette 54/68

JEE Mains 2018 MCQ

If $f\left( x \right) = \left| {\matrix{ {\cos x} & x & 1 \cr {2\sin x} & {{x^2}} & {2x} \cr {\tan x} & x & 1 \cr } } \right|,$ then $\mathop {\lim }\limits_{x \to 0} {{f'\left( x \right)} \over x}$

A

does not exist.

B

exists and is equal to 2.

C

existsand is equal to 0.

D

exists and is equal to $-$ 2.

Question 55: Palette 55/68

JEE Mains 2018 MCQ

If   x² + y² + sin y = 4, then the value of ${{{d^2}y} \over {d{x^2}}}$ at the point ($-$2,0) is :

A

$-$ 34

B

$-$ 32

C

4

D

$-$ 2

Question 56: Palette 56/68

JEE Mains 2017 MCQ

Let f be a polynomial function such that

f (3x) = f ' (x) . f '' (x), for all x $ \in $ R. Then :

A

f (2) + f ' (2) = 28

B

f '' (2) $-$ f ' (2) = 0

C

f '' (2) $-$ f (2) = 4

D

f (2) $-$ f ' (2) + f '' (2) = 10

Question 57: Palette 57/68

JEE Mains 2017 MCQ

If y = ${\left[ {x + \sqrt {{x^2} - 1} } \right]^{15}} + {\left[ {x - \sqrt {{x^2} - 1} } \right]^{15}},$

then (x² $-$ 1) ${{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$ is equal to :

A

125 y

B

124 y²

C

225 y²

D

225 y

Question 58: Palette 58/68

JEE Mains 2017 MCQ

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 $\sqrt x .g\left( x \right)$, then $g\left( x \right)$ equals

A

${{{3x\sqrt x } \over {1 - 9{x^3}}}}$

B

${{{3x} \over {1 - 9{x^3}}}}$

C

${{3 \over {1 + 9{x^3}}}}$

D

${{9 \over {1 + 9{x^3}}}}$

Question 59: Palette 59/68

JEE Mains 2014 MCQ

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:

A

${1 \over {1 + {{\left\{ {g\left( x \right)} \right\}}^5}}}$

B

$1 + {\left\{ {g\left( x \right)} \right\}^5}$

C

$1 + {x^5}$

D

$5{x^4}$

Question 60: Palette 60/68

JEE Mains 2013 MCQ

If $y = \sec \left( {{{\tan }^{ - 1}}x} \right),$ then ${{{dy} \over {dx}}}$ at $x=1$ is equal to :

A

${1 \over {\sqrt 2 }}$

B

${1 \over 2}$

C

$1$

D

$\sqrt 2 $

Question 61: Palette 61/68

JEE Mains 2011 MCQ

${{{d^2}x} \over {d{y^2}}}$ equals:

A

$ - {\left( {{{{d^2}y} \over {d{x^2}}}} \right)^{ - 1}}{\left( {{{dy} \over {dx}}} \right)^{ - 3}}$

B

${\left( {{{{d^2}y} \over {d{x^2}}}} \right)^{}}{\left( {{{dy} \over {dx}}} \right)^{ - 2}}$

C

$ - \left( {{{{d^2}y} \over {d{x^2}}}} \right){\left( {{{dy} \over {dx}}} \right)^{ - 3}}$

D

${\left( {{{{d^2}y} \over {d{x^2}}}} \right)^{ - 1}}$

Question 62: Palette 62/68

JEE Mains 2010 MCQ

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 $g\left( x \right) = {\left[ {f\left( {2f\left( x \right) + 2} \right)} \right]^2}$. Then $g'\left( 0 \right) = $

A

$-4$

B

$0$

C

$-2$

D

$4$

Question 63: Palette 63/68

JEE Mains 2009 MCQ

Let $y$ be an implicit function of $x$ defined by ${x^{2x}} - 2{x^x}\cot \,y - 1 = 0$. Then $y'(1)$ equals

A

$1$

B

$\log \,2$

C

$-\log \,2$

D

$-1$

Question 64: Palette 64/68

JEE Mains 2006 MCQ

If ${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$ then ${{{dy} \over {dx}}}$ is

A

${y \over x}$

B

${{x + y} \over {xy}}$

C

$xy$

D

${x \over y}$

Question 65: Palette 65/68

JEE Mains 2004 MCQ

If $x = {e^{y + {e^y} + {e^{y + .....\infty }}}}$ , $x > 0,$ then ${{{dy} \over {dx}}}$ is

A

${{1 + x} \over x}$

B

${1 \over x}$

C

${{1 - x} \over x}$

D

${x \over {1 + x}}$

Question 66: Palette 66/68

JEE Mains 2003 MCQ

If $f\left( x \right) = {x^n},$ then the value of

$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{f'''\left( 1 \right)} \over {3!}} + ..........{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \right)} \over {n!}}$ is

A

$1$

B

${{2^n}}$

C

${{2^n} - 1}$

D

$0$

Question 67: Palette 67/68

JEE Mains 2003 MCQ

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, $ then $f'\left( a \right),f'\left( b \right),f'\left( c \right)$ are in

A

Arithmetic -Geometric Progression

B

$A.P$

C

$G.P$

D

$H.P$

Question 68: Palette 68/68

JEE Mains 2002 MCQ

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

A

${n^2}y$

B

$-{n^2}y$

C

$-y$

D

$2{x^2}y$