Differentiation

If \(y=\sqrt{(x-\sin x)+\sqrt{(x-\sin x)+\sqrt{(x-\sin x) \ldots.}}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\)

Let \(\mathrm{P}(x)\) be a polynomial of degree 2, with \(\mathrm{P}(2)=-1, \mathrm{P}^{\prime}(2)=0, \mathrm{P}^{\prime \prime}(2)=2\), then \(\mathrm{P}(1.001)\) is

If \(\mathrm{f}^{\prime}(x)=\sin (\log x)\) and \(y=\mathrm{f}\left(\frac{2 x+3}{3-2 x}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=1\) is

If \(x=\sqrt{\mathrm{e}^{\sin ^{-1} t}}\) and \(y=\sqrt{\mathrm{e}^{\cos ^{-1} t}}\), then \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) is

Let \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) be a function such that \(\mathrm{f}(x)=x^3+x^2 \mathrm{f}^{\prime}(1)+x \mathrm{f}^{\prime \prime}(2)+6, x \in \mathrm{R}\), then \(\mathrm{f}(2)\) is

If \(y=\tan ^{-1}\left(\frac{4 \sin 2 x}{\cos 2 x-6 \sin ^2 x}\right)\), then \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)\) at \(x=0\) is

Differentiation of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)\) w.r.t. \(\cos ^{-1}\left(\sqrt{\frac{1+\sqrt{1+x^2}}{2 \sqrt{1+x^2}}}\right)\) is

If \(x=\log _e\left(\frac{\cos \frac{y}{2}-\sin \frac{y}{2}}{\cos \frac{y}{2}+\sin \frac{y}{2}}\right), \tan \frac{y}{2}=\sqrt{\frac{1-t}{1+t}}\) Then, \(\left(y_1\right)_{t=1 / 2}\) has the value

Let \(f: R \rightarrow R\) be a function such that \(f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R \text {, }\) then \(f(2)\) equals

\(y=\frac{\sqrt[3]{1+3 x} \sqrt[4]{1+4 x} \sqrt[5]{1+5 x}}{\sqrt[7]{1+7 x} \sqrt[8]{1+8 x}} \text {. Then, } \frac{d y}{d x} \text { at } x=0\) is

\(y=(1+x)\left(1+x^2\right)\left(1+x^4\right) \ldots \ldots \ldots\left(1+x^{2 n}\right)\),
then the value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

\(\text { If } \log (x+y)=2 x y \text {, then } \frac{\mathrm{d} y}{\mathrm{~d} x} \text { at } x=0 \text { is }\)

If \(x=-1\) and \(x=2\) are extreme points of \(\mathrm{f}(x)=\alpha \log x+\beta x^2+x, \alpha\) and \(\beta\) are constants, then the value of \(\alpha^2+2 \beta\) is

The derivative of \(\mathrm{f}(\tan x)\) w.r.t. \(\mathrm{g}(\sec x)\) at \(x=\frac{\pi}{4}\) where \(\mathrm{f}^{\prime}(1)=2\) and \(\mathrm{g}^{\prime}(\sqrt{2})=4\) is

The approximate value of \(\sin \left(60^{\circ} 0^{\prime} 10^{\prime \prime}\right)\) is (given that \(\sqrt{3}=1.732,1^{\circ}=0.0175^{\circ}\) )

If \(\tan y=\frac{x \sin \alpha}{1-x \cos \alpha}\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{m}}{x^2+2 \mathrm{n} x+1}\), then \(\mathrm{m}^2+\mathrm{n}^2\) is

For \(x>1\), if \((2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}\), then \(\left(1+\log _e 2 x\right)^2 \frac{d y}{d x}\) is equal to

If \(\mathrm{f}(x)=\sin ^{-1}\left(\frac{2 \log x}{1+(\log x)^2}\right)\), then \(\mathrm{f}^{\prime}(\mathrm{e})\) is

Derivative of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)\) w.r.t. \(\cos ^{-1} x^2\) is

Let \(\mathrm{f}(x)=\log (\sin x), 0 < x < \pi\) and \(\mathrm{g}(x)=\sin ^{-1}\left(\mathrm{e}^{-x}\right), x \geq 0\). If \(\alpha\) is a positive real number such that \(\mathrm{a}=(\mathrm{fog})^{\prime}(\alpha)\) and $$\mathrm{b}=(\mat...

If \(y=\log _{\sin x} \tan x\), then \(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{x=\frac{\pi}{4}}\) has the value

If \(y=[(x+1)(2 x+1)(3 x+1) \ldots(\mathrm{n} x+1)]^{\frac{3}{2}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=0\) is

The set of all points, where the derivative of the functions \(\mathrm{f}(x)=\frac{x}{1+|x|}\) exists, is

\(\text { For all real } x \text {, the minimum value of } \frac{1-x+x^2}{1+x+x^2} \text { is }\)

If \(\mathrm{f}(x)=3^x ; \mathrm{g}(x)=4^x\), then \(\frac{\mathrm{f}^{\prime}(0)-\mathrm{g}^{\prime}(0)}{1+\mathrm{f}^{\prime}(0) \mathrm{g}^{\prime}(0)}\) is

If \(\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x\) and \(\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))\), then \(\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}\) is

For \(x>1\), if \((2 x)^{2 y}=4 \mathrm{e}^{2 x-2 y}\), then \((1+\log 2 x)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to

If \(y=\cos ^{-1}\left(\frac{\mathrm{a}^2}{\sqrt{x^4+\mathrm{a}^4}}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is

If \(y=\tan ^{-1}\left(\frac{\log \left(\frac{\mathrm{e}}{x^2}\right)}{\log \left(e x^2\right)}\right)+\tan ^{-1}\left(\frac{4+2 \log x}{1-8 \log x}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is

If \(x=3 \tan \mathrm{t}\) and \(y=3 \sec \mathrm{t}\), then the value of \(\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}\) at \(\mathrm{t}=\frac{\pi}{4}\) is

If \(y\) is a function of \(x\) and \(\log (x+y)=2 x y\), then \(\frac{d y}{d x}\) at \(x=0\) is

Let \(f\) be a differentiable function such that \(\mathrm{f}(1)=2\) and \(\mathrm{f}^{\prime}(x)=\mathrm{f}(x)\), for all \(x \in \mathrm{R}\). If \(\mathrm{h}(x)=\mathrm{f}(\mathrm{f}(x))\), then \(\mathrm{h}^{\prime}(1)\) is equal to

If \(y=\sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right)\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) is equal to

If \(y=\log \sqrt{\frac{1+\sin x}{1-\sin x}}\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=\frac{\pi}{3}\) is

If \(y^{\frac{1}{m}}+y^{\frac{-1}{m}}=2 x, x \neq 1\), then \(\left(x^2-1\right)\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^2\) is equal to

\(\text { If } u=\cos ^3 x, v=\sin ^3 x \text {, then }\left(\frac{d v}{d u}\right)_{x=\frac{\pi}{4}} \text { is equal to }\)

If \(y=\log _{10} x+\log _x 10+\log _x x+\log _{10} 10\), then \(\frac{d y}{d x}=\)

If \(\sin ^2 x+\cos ^2 y=1\), then \(\frac{d y}{d x}=\)

If \(x=e^t(\sin t-\cos t)\) and \(y=e^t(\sin t+\cos t)\), then \(\frac{d y}{d x}\) at \(t=\frac{\pi}{3}\) is

If \(y=1+x e^y\), then \(\frac{d y}{d x}=\)

If \(f(x)=\operatorname{cosec}^{-1}\left[\frac{10}{6 \sin \left(2^x\right)-8 \cos \left(2^x\right)}\right]\), then \(f^{\prime}(x)=\)

If \(\mathrm{x}=\mathrm{a}\left(\mathrm{t}-\frac{1}{\mathrm{t}}\right)\) and \(\mathrm{y}=\mathrm{b}\left(\mathrm{t}+\frac{1}{\mathrm{t}}\right)\), then \(\frac{\mathrm{dy}}{\mathrm{dx}}=\)

If \(y=\log \sqrt{\tan x}\), then the value of \(\frac{d y}{d x}\) at \(x=\frac{\pi}{4}\) is

The derivative of the function \(\cot ^{-1}\left[(\cos 2 x)^{1 / 2}\right]\) at \(x=\pi / 6\) is

For all real \(x\), the minimum value of the function \(f(x)=\frac{1-x+x^2}{1+x+x^2}\) is

If \(y=x \tan y\), then \(\frac{d y}{d x}=\)

\(x=\frac{1-t^2}{1+t^2}\) and \(y=\frac{2 a t}{1+t^2}\), then \(\frac{d y}{d x}=\)

If \(y^2=a x^2+b x+c\), where \(a, b, c\) are constants, then \(y^3 \frac{d^2 y}{d x^2}\) is equal to

If \(x^y \cdot y^x=16\), then $\frac{d y}{d x}$ at $(2,2)$$ is

If \(y=\tan ^{-1} \sqrt{\frac{1+\cos x}{1-\cos x}}\), then \(\frac{d y}{d x}=\)