Differentiability Questions - JEE

If f : ℝ → ℝ is the function defined by f(x) = (ex − e−x)/(ex + e−x), then

Let $f:[−1,3]\to\mathbb{R}$ be defined as $f(x)=\begin{cases}|x|+[x], & -1\leq x where \([t]$ denotes the greatest integer less than or equal to $t$. Then $f$ is discontinuous at

Let $f(x) = \int_0^x t\ln(1+t^2)\, dt$, then $f''(0)$ is:

If $2^x + 2^y = 2^{x+y}$, then $\dfrac{dy}{dx}$ is equal to which of the following expressions?

Let f : R → R be a continuous function satisfying $ f(x) + \int_0^x t f(t)\, dt + x^2 = 0 $ ∀ x. Then:

Let $h(x) = \min\{x, x^2\}$ for every real number $x$. Then,

Let \[S = \{(\lambda, \mu) \in R \times R : f(t) = (|\lambda|e^{|t|} - \mu)\sin(2|t|),\, t \in R,\] is a differentiable function\}. Then $S$ is a subset of

If $f(x) = \begin{cases} 2x^2 + 3 & x > 3 \\ ax^2 + bx + 1 & x \leq 3 \end{cases}$ is differentiable everywhere, then $a$ is equal to

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 $k$.]

The function $f(x) = \max\{(1-x), (1+x), 2\}$ $\forall x \in \mathbb{R}$ is

If $y = \tan^{-1}\left(\frac{1}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ ($y > 0$), then $\frac{dy}{dx}$ is equal to

If $a = \sec f$ and $b = \csc f$, where $f$ is a parameter, then the value of $\frac{dy}{dx}$ when $x = -\frac{3}{4}$ is

If $y = (1+x^2)^n + \sin^{-1}(\sin^2 x)$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is

Let $f(x) = x\left[\frac{x}{2}\right]$, for \(-10

Let h(x) = 2 − |x − 1| and g(x) = h(|x|) + |h(x)|. Find the number of points where g(x) is non-differentiable.

The value of $\lim_{x \to \infty} \frac{ax^2 + bx + c}{dx + e}$ ($a, b, c, d, e \in \mathbb{R} - \{0\}$) depends on the sign of:

If $f(x) = 2x^3 - 3x^2 + a$ is a one-to-one function, the value of $a$ is arbitrary. Given $f(g(x))=x$, then $g'\!\left(\dfrac{11}{4}\right)$ is:

\[ \lim_{x \to 0} \frac{\sin{(\pi \cos{x})}}{x^2} \]

Find $\frac{dy}{dx}$ at $x = 3$ where $y = \sqrt{x^2 + 16}$ and $s = \frac{x}{y}$

If $x = \frac{1-\tan^2\theta}{1+\tan^2\theta}$ and $y = \frac{2\tan\theta}{1+\tan^2\theta}$, find $\frac{dy}{dx}$

25. If $y = (x + \sqrt{1 + x^2})^n$, then the value of $(1 + x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}$ is equal to:

Consider the function $f(x) = \tan^{-1}\left(\frac{x^2-1}{x^2+1}\right), \forall x \geq 0$. If $g(x)$ is the inverse function of $f(x)$, then the value of $g'\left(\frac{\pi}{4}\right)$ is equal to

The integer $n$ for which $\lim_{x \to 0} (\sin x)^{1/x}$ is a finite non-zero number, is [2002 AIEEE]

If $I_n = \frac{d^n}{dx^n}(x^n\ln x)$, then the value of $\frac{d}{dx}(I_7 - 7I_6)$ is equal to

198. If $y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}$, then the value of $\dfrac{dy}{dx}$ at $x = 8$ is:

The total number of points of non-differentiability of $f(x) = \max\left\{\sin^2 x, \cos^2 x, \frac{3}{4}\right\}$ in $[0, 10\pi]$, is

If $\lim_{x \to \frac{\pi}{2}} \frac{\sin x}{x}$ exists finitely, then the value of $a$ is

The value of $\lim_{x \to \pi} \left[1 - \cos\left(\frac{x}{2}\right) - \cos\left(\frac{x}{4}\right) + \cos\left(\frac{x}{2}\right) \cdot \cos\left(\frac{x}{4}\right)\right]$ is $\frac{\lambda}{4}$, then the value of $900\lambda$ is equal to there, $\lambda > 0$)

If $\lim_{x \to 0} \frac{\sin 3x}{\sin 4x} - 3$, then the value of $272 \cdot \frac{ab}{cd}$ is equal to

The value of $\lim_{x \to 0} \frac{\sin\left(\frac{x}{3}\right) \sin\left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}$ is equal to

If $x = t^3 + t^5$ and $y = \sin t$, then $\dfrac{d^2y}{dx^2}$ equals:

Let $f(x) = x^2 - 4x - 3, x > 2$ and $g(x)$ is the inverse of $f(x)$. Then the value of $\frac{1}{g'(2)}$, where $f(z) = 2$ is (here, $g'$ represents the first derivative of $g$)

Find $\displaystyle\lim_{x \to \infty} \frac{\displaystyle\int_0^x e^{t^2}\, dt}{\displaystyle\int_0^x e^{2t^2}\, dt}$.

198. If $y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}$, then the value of $\dfrac{dy}{dx}$ at $x = 8$ is:

Find $\lim_{x \to 4} \frac{(1-\sin(\pi x))}{(x-4)^2}$

If $y = 1 + \dfrac{c_1}{x - c_1} + \dfrac{c_2 x}{(x - c_1)(x - c_2)} + \dfrac{c_3 x^2}{(x - c_1)(x - c_2)(x - c_3)}$, then $\dfrac{dy}{dx}$ is equal to:

Let $f$ be continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a)/a = f(b)/b$. Then by Rolle's theorem applied to $g(x) = f(x)/x$, there exists $x_0 \in (a, b)$ such that:

196. Let f and g be defined such that $f'(x) = f^2(x) + g^2(x)$ and $g'(x) = 2f(x)g(x) + 1$. If $f(0) = \dfrac{1}{5}$, $g(0) = \dfrac{4}{5}$, then the value of $f\!\left(\dfrac{\pi}{12}\right) + g\!\left(\dfrac{\pi}{12}\right)$ equals:

Consider the following statements about positive functions $f(x)$ and $g(x)$ whose limits to infinity exist.Statement-A: $\lim_{x\to\infty} f(x) = \lim_{x\to\infty} g(x)$Statement-B: $\lim_{x\to\infty}(f(x)-g(x))=0$Statement-C: $\lim_{x\to\infty}\sqrt{f(x)} = \lim_{x\to\infty}\sqrt{g(x)}$How many of the following six statements are true: $A\Rightarrow B,\; B\Rightarrow C,\; C\Rightarrow A,\; A\Rightarrow C,\; B\Rightarrow A,\; C\Rightarrow B$?

The value of $$\lim_{x \to 0^+} \frac{\int_1^{\cos x} (\cos^{-1} t) \, dt}{2x - \sin 2x}$$ is equal to:

If $x = e^t \cos t$, $y = e^t \sin t$ then $\dfrac{d^2 y}{dx^2}$ = ______

Let p = $ \lim_{x \to 0^+} (1 + \tan^2{\sqrt{x}})^{\frac{1}{x}} $ then log p is equal to -

Let $f(x) = x \cos x + 2$ and $g(x)$ be the inverse function of $f(x)$, then $g'(3)$ equals to ……… .

If $\displaystyle\lim_{\alpha\to 0} \frac{e^{\cos(\alpha^n)} - e}{\alpha^m} = \frac{-e}{2}$ where $m$ and $n$ are positive integers greater than 1, then the value of $\dfrac{m}{n}$ is:

$\lim_{x \to 0} \dfrac{x \cot(4x)}{\sin^2 x \cot^2(2x)}$ is equal to __________.

797. Let $f(x) = x$, $g(x) = |1 - f(x)|$, $h(x) = 2 - g(x)$, $L(x) = h(|x|) + |h(x)|$. Find the number of points where $L(x)$ is non-differentiable.

The number of values of $x$, $x \in [-2, 3]$ where $f(x) = [x^2]\sin(\pi x)$ is discontinuous is (where $[\cdot]$ denotes greatest integer function)

If f(x) = $\begin{cases} \frac{-x}{2}, & x > 0 \\ -\cos x, & 0 \leq x \leq 1 \\ \ln x, & x > 1 \end{cases}$, then find the number of points where f(x) is not differentiable.

$\lim_{x \to 0} \dfrac{x + 2\sin x}{\sqrt{x^2 + 2\sin x + 1} - \sqrt{\sin^2 x - x + 1}}$ is __________.

If $ f(x) = \begin{cases} ax + b, & x \le -1 \\ ax^3 + x + 2b, & x > -1 \end{cases} $ is differentiable for all $ x \in \mathbb{R} $, then find $ b - a = $ __________.

Differentiability Questions (1063)