Differentiability Questions - JEE

If $f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$, then $$\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}, \quad 0

If $x + |y| = 2y$ then $y$ as a function of $x$ is

197. Let $g(x) = \dfrac{1}{f^{-1}(x)}$. Given the following data:$x$01234$f(x)$$-2$$-1$246$f'(x)$1/22/314/35/3The value of $g'(4)$ is:

If $f(x) = \begin{vmatrix} x & x^2 & x^3 \\ 1 & 2x & 3x^2 \\ 0 & 2 & 6x\end{vmatrix}$, find $f'(1)$.

Let $f: R \to R$ be a function defined as \[f(x) = \begin{cases} 5, & \text{if } x \leq 1 \\ a + bx, & \text{if } 1

If the function \[f(x) = \begin{cases} x^3, & -2 \leq x where $f(x) = \sin(x-2) + a\cos(x-2)$is continuous and differentiable in $(4, 6)$, then find the range of $a$.

Let f(x) = P10 r=1 r sin rx x . The value of limx→0 f(x) is:

Let $f(x) = \dfrac{\tan x}{x}$, then the value of $\lim_{x \to 0}\left([f(x)] + x^2\right)^{\frac{1}{\{f(x)\}}}$ is equal to(where [.] and {.} denote greatest integer function and fractional part functions respectively)

Given limit = $\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{f(\sec^2 x) \cdot 2\sec^2 x \tan x}{2x} = \dfrac{8}{\pi} f(2)$. If $k^a = 8^2$, find the value of $k^a$.

$f(x)=4|2x+3|+9[x+\frac{1}{2}]-12[x+20]$. Non-diff in (-20,20):

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\dfrac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}$, $x\neq0$. If $L$ and $R$ respectively denote the left hand limit and the right hand limit of $f(x)$ at $x=0$, then $\dfrac{32}{\pi^2}(L^2+R^2)$ is equal to

The value of limn→∞(4n + 5n)1/n is:

55. If \(f(x) = \begin{cases} -e^{-x} + k, & x \leq 0 \\ e^x + 1, & 0

lim n→∞tan n X r=1 tan−1 1 1 + r + r2 ! is equal to:

Let $f(x) = x^3 - 9x^2 + 24x - 4$. Let $g(x)$ be defined as follows: \[g(x) = \begin{cases} f(x+2); & x If $g(x)$ is continuous for all $x$, find the value of $a$.

Let $u(x)$ and $v(x)$ be differentiable functions such that $\dfrac{u(x)}{v(x)} = 7$. If $\dfrac{u'(x)}{v'(x)} = p$ and $\left(\dfrac{u(x)}{v(x)}\right)' = q$, then $\dfrac{p+q}{p-q}$ has the value equal to:

30. Let $y$ be an implicit function of $x$ defined by $x^{2x} - 2x^x \cot y - 1 = 0$. The value of $y'(1)$, where $y'$ denotes the first derivative of $y$, is:

If the function $f(x)=\begin{cases}(1+|\cos x|)^{\frac{\lambda}{|\cos x|}}, & 0<x<\dfrac{\pi}{2}\\\mu, & x=\dfrac{\pi}{2}\\\dfrac{\cot 6x}{e^{\cot 4x}}, & \dfrac{\pi}{2}<x<\pi\end{cases}$ is continuous at $x=\dfrac{\pi}{2}$, then $9\lambda+6\log_e\mu+\mu^6-e^{6\lambda}$ is equal to:

Let $f(x) = \begin{cases} \left[1 + \ln(c^2 + c + 1)\tan^2(x-1)\right]^{\frac{1}{(\ln x)^2}}, & x \neq 1 \\ 3c, & x = 1 \end{cases}$, where $c \in R$.If $\lim_{x \to 1} f(x)$ exists but $f(x)$ is discontinuous at $x = 1$, then $c$ can take the value:

The function $f(x) = |x-3|, x \geq 1$\(\frac{x^2}{4} - \frac{3x}{2} + \frac{13}{4}, x

199. Let $f(x)$ be a function defined by $f(x) = (k - x^{10})^{1/10}$ where $k = 1025$ and $f'(2) = \dfrac{1}{f'(a)}$ where $a \in N$, then $a$ equals:

The value of $\lim_{x \to 0} \left[\frac{3}{2} + \frac{x^2}{\sin x \tan x}\right]$ (where $[\cdot]$ denotes the greatest integer function) is

If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{\pi}{4}$ is:

For any positive integer n, define fn(x) = n X j=1 tan−1 1 1 + (x + j)(x + j −1) , x ∈(0, ∞). Then which of the following statement(s) is(are) true?

The value of limx→−1 cos 2−cos 2x x2−|x| is:

252. The number of points where $f(x) = |x + [x]| - 3[2x] + 4[3x]$ is discontinuous in $[-1, 1]$, is:[Note: $[k]$ denotes greatest integer less than or equal to $k$.]

If β = limx→0 ex3−(1−x3)1/3+((1−x2)1/2−1) sin x x sin2 x , find 6β.

Let $y(x)=\left(1+x\right)\!\left(1+x^2\right)\!\left(1+x^4\right)\!\left(1+x^8\right)\!\left(1+x^{16}\right)$. Then $y'(1)-y(1)$ is:

If $3f(x) - 2f(1/x) = x$, then $f'(2)$ is equal to:

For n ∈N, let an = Pn k=1 2k and bn = Pn k=1(2k −1). Then limn→∞(√an −√bn) is:

Let f: \mathbb{R}^+ \to \mathbb{R} be a differentiable function satisfying: \frac{f(x)}{y} + \frac{f(y)}{x} + f(xy) = \frac{}{} for all x, y \in \mathbb{R}^+, with f(1) = 0 and f'(1) = 1. Find \lim_{x \to e} \lfloor f(x) \rfloor (where \lfloor \cdot \rfloor denotes greatest integer function).

Let α(a) and β(a) be the roots of the equation ( 3√1 + a−1)x2+(√1 + a−1)x+( 6√1 + a−1) = 0. Find lima→0+ α(a) and lima→0+ β(a).

If x = \cos\theta and y = \sin^3\theta, then find y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 at \theta = \frac{\pi}{2}.

Let $f$ be a continuous function on $\mathbb{R}$ such that $f\left(\frac{1}{n^4}\right) = \frac{(\sin e^{-n^2})e^{-n^2} + \frac{n^2}{n^2+1}}{1}$. Then $f(0) = $

If $f(x) = \cot^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\frac{1 - x^2}{1 + x^2}\right)$ then $\lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)}$ for \(0

Let $S$ be the set of all points in $(-\pi,\pi)$ at which the function $f(x)=\min(\sin x,\cos x)$ is not differentiable. Then $S$ is a subset of which of the following?

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 $y\!\left(\tfrac{1}{2}\right)=-\tfrac{1}{4}$. Then $\dfrac{dy}{dx}$ at $x=\tfrac{1}{2}$:

Let $f$ be defined for all $x \in \mathbb{R}$. If $f$ is differentiable and $f(x^3) = x^5$ for all $x \in \mathbb{R}\,(x\ne0)$, then $f'(27)$ is equal to:

If $y = f(x)$ satisfies $f(x+y) = f(x)+f(y)+2xy-1$ for all $x,y\in\mathbb{R}$ and $f'(0) = \cos\alpha$, then which of the following is/are true?

$h(x)=\min\{\{x\},1-\{x\}\}$. h is:

Let $f$, $g$ and $h$ be differentiable functions. If $f(0) = 1$, $g(0) = 2$, $h(0) = 3$ and the derivative of their pairwise product at $x = 0$ are $(fg)'(0) = 6$, $(gh)'(0) = 4$ and $(hf)'(0) = 5$, then compute the value of $(fgh)'(0)$.

771. Let $f: R \to R$ be a function defined by $f(x) = \begin{cases} x^3 + 2x^2 + x + c, & \text{if } x \leq b \\ e^x, & \text{if } x > b \end{cases}$, where $b$ and $c$ are integers. If $f(x)$ is differentiable $\forall\, x \in R$, then find the value of $(b + c)$.

Let $f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ where $a, b, c, d$ are in A.P. If $f(x)$ is continuous, then the number of local minima of $y = |f(|x|)|$ is:

Let \(g(x) = \begin{cases} k\sqrt{x+1}; & 0 \leq x \leq 3 \\ mx + 2; & 3

A certain function $f(x)$ has the property that $f(3x) = af(x)$ for all positive real values of $x$ and $f(x) = 1 - |x - 2|$ for $1 \le x \le 3$. Find $\lim_{x \to 2} (f(x))^{\frac{\csc\left(\frac{\pi x}{2}\right)}{1}}$

If $f(x) = \begin{cases} x\tan^{-1}(1/x), & x \in (-\infty,-1]\cup[1,\infty) \\ 0, & x = 0 \end{cases}$, then $f'(0)$ is:

If \(\alpha,\ \beta,\ (\alpha

Question 23: Given We have f: (-1, 1) → ℝf(0) = -1, f'(0) = 1If g(x) = [f(2f(x) + 2)]^(1/2)Find g'(0)

The value of limx→0(1 + sin x)cos x is:

Evaluate: $\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{\sin x}{x - \sin x}}$

Differentiability Questions (1063)