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

If $x+y=\tan ^{-1} y$ and $\frac{d^{2} y}{d x^{2}}=f(y) \frac{d y}{d x}$,then $f(y)$ is equal to

If ${x^3} + 8xy + {y^3} = 64$,then $\frac{dy}{dx} = $

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \,d t=\int_{0}^{x} f(t) \,d t$ for $0 \leq x \leq 1$ and $f(0)=0$,then $\lim_{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) \,d t$ is:

If ${x^{2/3}} + {y^{2/3}} = {a^{2/3}}$,then $\frac{dy}{dx} = $

Let $y$ be an implicit function of $x$ defined by ${x^{2x}} - 2{x^x}\cot y - 1 = 0$. Then $y'(1)$ equals