If $\tan^{-1}(x^2 + y^2) = \alpha$,then $\frac{dy}{dx}$ is equal to

For $x \in R$,$f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$,then

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

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

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

Let $f : (0, \infty) \rightarrow (0, \infty)$ be a differentiable function such that $f(1) = e$ and $\lim_{t \rightarrow x} \frac{t^{2} f^{2}(x) - x^{2} f^{2}(t)}{t - x} = 0$. If $f(x) = 1$,then $x$ is equal to