If $y = \sqrt[3]{\tan x + y}$, then $\frac{dy}{dx} =$

If $e^y + xy = e$, then the ordered pair $(\frac{dy}{dx}, \frac{d^2y}{dx^2})$ at $x = 0$ is equal to

If $x > 0$ and $x^y = e^{x-y}$, then $\frac{dy}{dx}$ is equal to

The equation of the normal to the curve $y=(1+x)^{2y}+\cos^{2}(\sin^{-1} x)$ at $x=0$ is

The focal length of a mirror is given by $\frac{2}{f} = \frac{1}{v} - \frac{1}{u}$. In finding the values of $u$ and $v$, the errors are equal to $p$. Then, the relative error in $f$ is

If $\log (x+y)-2xy=0$,then $y^{\prime}(0)=$