If $\sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 6$, then $\frac{dy}{dx} = $

If $(x^2-3x+2) e^{\frac{y}{x-1}}=x+2$, then find the value of $(\frac{dy}{dx})_{x=0}$.

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two non-constant differentiable functions. If $f^{\prime}(x) = e^{(f(x)-g(x))} g^{\prime}(x)$ for all $x \in R$,and $f(1) = g(2) = 1$,then which of the following statement$(s)$ is (are) $TRUE$?

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

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 $\frac{dy}{dx}=$