If $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4$,then $\frac{d y}{d x}=$

If $x^2+y^2=t+\frac{1}{t}$ and $x^4+y^4=t^2+\frac{1}{t^2}$,then $x^3 y \frac{dy}{dx}$ is equal to

If $x e^{xy} = y + \sin^2 x$, then the value of $\frac{dy}{dx}$ at $x = 0$ is equal to

If $x \sqrt{1+y}+y \sqrt{1+x}=0$,then $\frac{d y}{d x}$ is equal to

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