List of top Mathematics Questions on Differentiation asked in TS EAMCET

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

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

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

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

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

Let f and g be two differentiable functions satisfying $g'(5) = \frac{3}{4}$, $g(5) = 6$ and $g = f^{-1}$. Then $f'(6) =$

If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2y}{dx^2}=-1$ at $t=K, K>0$, then $\lim_{t \to K} \frac{y}{x} =$

If $y=f(x)^{g(x)}$ and $\frac{dy}{dx} = y[H(x)f'(x)+G(x)g'(x)]$, then $\int \frac{G(x)H(x)f'(x)}{g(x)}dx =$

The set of all values of x for which $f(x) = ||x|-1|$ is differentiable is

If \( f(x) = \begin{cases} x^2 \cos\left(\frac{\pi}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases} \), then at \( x = 0 \), \( f(x) \) is

If $x = \sin 2\theta \cos 3\theta$, $y = \sin 3\theta \cos 2\theta$, then $\frac{dy}{dx} =$

If $x = \sqrt{1-\tan y}$, then $\frac{dy}{dx} =$

If $y = \sqrt{\log(x^2+1)+\sqrt{\log(x^2+1)+\sqrt{\log(x^2+1)+...}}}$, $|x|<1$, then $\frac{dy}{dx} =$

If $f(x) = \log_{(x-1)^2}(x^2-3x+2)$, $x \in \mathbb{R}-[1,2]$ and $x\neq0$, then $f'(3)=$

If $y = \text{Sec}^{-1}x$, then $\frac{d^2y}{dx^2} =$

If $y=(1-x^2)\text{Tanh}^{-1}x$ then $\frac{d^2y}{dx^2}=$

The values of x at which the real valued function $f(x)=7|2x+1|-19|3x-5|$ is not differentiable is

If $y^3=x$ then the value of $\frac{dy}{dx}$ at $x=1$ is