List of top Mathematics Questions on Differentiation

$[\frac{d}{dx}((sin~x)^{cos~x})]_{x=7/4}=$}

The number of critical points of the function} \[ f(x)= \begin{cases} \dfrac{|\sin x|}{x}, & x\neq0\\ 1, & x=0 \end{cases} \] in the interval \((-2\pi,2\pi)\) is equal to:

If \[ y=(e^{2x}-4)(6e^{2x}-5e^{x}+1), \] then \[ \left(\frac{dy}{dx}\right)_{x=0} - \left(\frac{d^{2}y}{dx^{2}}\right)_{x=0} = \ ? \]

If \(f(x)\) is a differentiable function and \[ y=e^{f(x)+e^{f(x)+e^{f(x)+\cdots \infty}}}, \] then \[ \frac{dy}{dx}= \ ? \]

If \[ \frac{d}{dx} \left( \frac{\sec x+\tan x} {\sec x-\tan x} \right) =k \] at \[ x=\frac{\pi}{4}, \] then \[ \frac{k}{2\sqrt2}-2\sqrt2= \ ?} \]

\(f(x)\) is an \(n^{th}\) degree polynomial and \(\alpha_1,\alpha_2,\ldots,\alpha_n\) are distinct zeros of \(f(x)\). \(g(x)\) is a polynomial having three zeros common with the zeros of \(f(x)\). Assertion (A): \(|f(x)|g(x)\) is continuous and differentiable at all \(\alpha_i\).
Reason (R): \[ \lim_{x\to a}\frac{|x-a|}{x-a} \] does not exist and \[ \lim_{x\to a}|x-a|=0. \] Choose the correct option.}

If \[ y=\sec^{-1}\left(\frac{1+x^2}{2x}\right) \quad \text{and} \quad x>1, \] then \[ \frac{dy}{dx}= \]

If \[ f(x)=\sqrt{2^{2x}\log(3x-2)} \] then \(f'(2)=\)

If \[ (3y)^{2x}=5\left(2^{3x}\right), \] then \[ \left(\frac{dy}{dx}\right)_{x=1} = \]

If \[ y=\tan^{-1}\left[\left(\frac{1-\cos 2\sqrt{x}}{1+\cos 2\sqrt{x}}\right)^{\frac12}\right], \qquad 0<x<\frac{\pi^2}{4}, \] then \(y(2y'+y)=\)

If \(f:\mathbb{R}-\{0\}\rightarrow\mathbb{R}\) is a differentiable function such that \[ \frac{1}{3}f(x)+3f\!\left(\frac1x\right) = x-\frac{10}{3}, \] then find \[ f'(3)-f'\!\left(\frac13\right). \]

If a function \[ f(x)= \begin{cases} \dfrac{a}{|x|}, & x\le -1 \text{ or } x\ge 1,[6pt] \\ x^2+b, & -1<x<1, \end{cases} \] is differentiable on \(\mathbb{R}\), then \(a+b=\)

If \[ f(x)=(1+x^3)(1+x^6)(1+x^{12})(1+x^{24}) \] then \(f'(-1)=\)

If \[ f(x)=\pi-\cos^{-1}\left(\frac{x^2+4x+3}{x^2+4x+5}\right) \] then \(f'(1)=\)

If \(x \in [-1,1]\) and \(y=(\cot^{-1}x)^{\cot^{-1}x}\), then \(\left(\frac{dy}{dx}\right)_{x=0}=\)

If \(f(x)=\sqrt{-(1+x)}\sec^{-1}x\) is a real valued function, then \(f'(x)=\)

If \(x=\sinh^{-1}t+\log(t^{2}+1)\) and \(y=\tan^{-1}t+\log|t|\), then \(\frac{dy}{dx}=\)

If \( x^{2}y - xy^{2} + x^{3} - y^{3} = 0 \), then \( \frac{dy}{dx} \) at the point (1, 1) is

If \( y=\text{sech}^{-1}\left(\frac{9}{9x^{2}+10}\right) \), then \( \frac{dy}{dx} = \)

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbb{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ is equal to :

Let \( f(x) \) be a polynomial of degree 5, and have extrema at \( x = 1 \) and \( x = -1 \). If \( \lim_{x \to 0} \frac{f(x)}{x^3} = -5 \), then \( f(2) - f(-2) \) is equal to:

The number of points, at which the function \(f(x) = \max\{6x, 2 + 3x^2\} + |x - 1| \cos|x^2 - \frac{1}{4}|\), \(x \in (-\pi, \pi)\), is not differentiable, is ____.

Let \(f\) be a real polynomial of degree \(n\) such that \(f(x) = f'(x)f''(x)\), for all \(x \in \mathbb{R}\). If \(f(0) = 0\), then \(36(f''(2) + f''(2) + \int_0^2 f(x)\,dx)\) is equal to:

If $f(x)=\begin{cases}\frac{1-\sin^{3}x}{3\cos^{2}x}, & x\ne\frac{\pi}{2} \\ \frac{1}{2}, & x=\frac{\pi}{2}\end{cases}$, then $f^{\prime}\left(\frac{\pi}{2}\right)=$

If $\tan^{-1}x^{2}+\tan^{-1}y^{2}=\frac{\pi}{2}$, then $\left(\frac{dy}{dx}\right)_{(-1,2)}=$