If $\sin ^2 x + \cos ^2 y = 1$,then $\frac{dy}{dx} = $

If $2^x+2^y=2^{x+y}$, then $\frac{d y}{d x}=$

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 $x^{p}+y^{q}=(x+y)^{p+q}$,then $\frac{dy}{dx}$ is

If $xy \neq 0, x+y \neq 0$ and $x^m y^n=(x+y)^{m+n}$,where $m, n \notin N$,then $\frac{dy}{dx}$ is equal to

If $ x^{y}=e^{x-y} $ then $ \frac{d y}{d x} $ is equal to