If $\ln \left( {(e - 1){e^{xy}} + {x^2}} \right) = {x^2} + {y^2}$,then $\left. {\frac{{dy}}{{dx}}} \right|_{(1,0)}$ is

If $x^{2}+y^{2}=t+\frac{1}{t}$ and $x^{4}+y^{4}=t^{2}+\frac{1}{t^{2}}$,then find $\frac{d y}{d x}$.

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 $f(x) = \frac{1}{x^3} \int_5^x (2u^2 - u f'(u)) du$,then $f'(5) = $

If ${x^2} + {y^2} = 1$,then find the relation between $y'$ and $y''$,where $y' = \frac{dy}{dx}$ and $y'' = \frac{d^2y}{dx^2}$.

If $x^2 \tan ^{-1} \frac{y}{x}-y^2 \tan ^{-1} \frac{x}{y}=k$,then $\left(\frac{d y}{d x}\right)_{(1,1)}=$