If $\frac{y}{x} \cos^4 \alpha + \frac{x}{y} \sin^4 \alpha = 2 \sin^2 \alpha \cos^2 \alpha$,then $\frac{dy}{dx} = $

If $y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + \dots \infty}}}$,then $\frac{dy}{dx} = $

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} = $

Find the derivative of $e^{x}+e^{y}=e^{x+y}$.

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