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

If $\cos ^{-1}\left(\frac{y}{b}\right)=n \log \left(\frac{x}{n}\right)$,then

If $x^y=y^x$,then $x(x-y \log x) \frac{d y}{d x}$ is equal to :

If $y$ is a function of $x$ and $\log (x+y)=2xy$,then the value of $y^{\prime}(0)$ 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 $x^2 y^2 = \sin^{-1} x + \cos^{-1} x$,then $\frac{dy}{dx}$ at $x = 1$ and $y = 2$ is