Differentiation

If $y=|\cos x-\sin x|+|\tan x-\cot x|$, then $\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{3}}+\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{6}}=$

If $\cos ^{-1}\left(\frac{y}{b}\right)=2 \log \left(\frac{x}{2}\right)$, where $x>0$, then $x^2 \frac{d^2 y}{d x^2}+x \frac{d y}{d x}$ is equal to

If $y=\operatorname{Tan}^{-1}\left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right)$ for $0 < |x| < 1$, then $\frac{d y}{d x}=$

If $f^{"}(x)$is continuous at $x=0$and $f^{"}(0)=4$, then find the following value of$\underset{x\rightarrow\,0}{\lim\,}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^{2}}$

If $y=a \cos (\log x)+b \sin (\log x)$, where $a, b$ are parameters, then $x^2 y^{\prime \prime}+x y^{\prime}$ is equal to

If \(y=f(x)\) is twice differentiable function such that at a point \(P, \frac{d y}{d x}=4, \frac{d^2 y}{d x^2}=-3\), then \(\left(\frac{d^2 x}{d y^2}\right)_P=\)

$f(x)$ and $g(x)$ are differentiable functions such that $\frac{f(x)}{g(x)}=$ a non zero constant. If $\frac{\mathrm{f}^{\prime}(\mathrm{x})}{\mathrm{g}^{\prime}(\mathrm{x})}=\alpha(\mathrm{x})$ and…

The derivative of \(f(x)=\cos ^{-1}\left[\sin \sqrt{\frac{1+x}{2}}\right]+x^x\) with respect to \(x\) at \(x=1\) is equal to

. Displacement ' $s$ ' of a particle, in meters, at any time ' $t$ ' $s$ in expressed as $s=\frac{t^3}{3}-6 t$. Find the acceleration at a time when the velocity vanishes,

If $x=\sec\,\theta\,-\cos\,\theta\,$and $y=\sec\,^{n}\theta\,-\cos\,^{n}\theta\,$, then $\left(x^{2}+4\right)\left(\frac{dy}{dx}\right)^{2}=$

Let $\quad f(x)=e^x, \quad g(x)=\sin ^{-1} x \quad$ and $h(x)=f(g(x))$, then $\frac{h^{\prime}(x)}{h(x)}$ is equal to

If $y_k$ is the $k$ th derivative of $y$ with respect to ' $x$ ', and $y=\cos (\sin x)$, then $y_1 \sin x+y_2 \cos x$ is equal to