A function $f: R \rightarrow R$ is such that $y f(x+y)+\cos m x y=1+y f(x)$. If $m=2$, then $f^{\prime}(x)=$

If $y=\sqrt{\sin (\log 2 x)+\sqrt{\sin (\log 2 x)+\sqrt{\sin (\log 2 x)+\ldots \infty,}}}$ then $\frac{d y}{d x}=$

If $y=\tan ^{-1}\left[\frac{\sin ^{3}(2 x)-3 x^{2} \sin (2 x)}{3 x \sin ^{2}(2 x)-x^{3}}\right]$, then $\frac{d y}{d x}=$

Derivative of $(\sin x)^{x}$ with respect to $x^{(\sin x)}$ is

If $y=\log \left(x-\sqrt{x^{2}-1}\right)$, then $\left(x^{2}-1\right) y^{\prime \prime}+x y^{\prime}+e^{y}+\sqrt{x^{2}-1}=$

If $y=\log \left[\tan \sqrt{\frac{2^x-1}{2^x+1}}\right], x>0$, then $\left(\frac{d y}{d x}\right)_{x=1}=$

If $\log y=y^{\log x}$, then $\frac{d y}{d x}=$

If $y=a \cos 3 x+b e^{-x}$, then $y^{\prime \prime}(3 \sin 3 x-\cos 3 x)=$

If $y=\frac{\tan x \cos ^{-1} x}{\sqrt{1-x^2}}$, then the value of $\frac{d y}{d x}$, when $x=0$ is

If $y(\cos x)^{\sin x}=(\sin x)^{\sin x}$, then the value of $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$ is

If $y=44 x^{45}+45 x^{-44}$, then $y^n=$

If $2 x^2-3 x y+4 y^2+2 x-3 y+4=0$, then $\left(\frac{d y}{d x}\right)_{(3,2)}=$

If $y=\sin a x+\cos b x$, then $y^{\prime \prime}+b^2 y=$

A particle moving from a fixed point on a straight line travels a distance $S$ metres in $t \mathrm{sec}$. If $S=t^3-t^2-t+3$, then the distance (in mts) travelled by the particle when it comes to rest, is

If $\tan y=\cot \left(\frac{\pi}{4}-x\right)$, then $\frac{d y}{d x}=$

If $x=3 \sqrt{2} \cos ^3 \theta$ and $y=4 \tan ^2 \theta$, then $\left(\frac{d y}{d x}\right)_{\theta=\pi / 4}=$

The derivative of $\frac{1-x^2}{1+x^2}$ with respect to $\frac{2 x}{1+x^2}$ at $x=2$ is

If the slope of the tangent drawn to the curve $y=e^{a+b x^2}$ at the point $P(1,1)$ is -2 , then the value of $2 a-3 b$ is