If $\log (x+y)=\sin (x+y)$,then $\frac{dy}{dx}$ is

If $x^y = e^{x - y}$,then $\frac{dy}{dx}$ at $x = 1$ is . . . . . .

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

If $\sqrt{\frac{y}{x}}+\sqrt{\frac{x}{y}}=2$, then $\frac{dy}{dx}$ is equal to

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