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

If $y=x \tan y$,then $\frac{d y}{d x}=$

Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1) .$ If $\int_{0}^{x} \sqrt{1-\left(f^{\prime}(t)\right)^{2}} \,d t=\int_{0}^{x} f(t) \,d t$ for $0 \leq x \leq 1$ and $f(0)=0$,then $\lim_{x \rightarrow 0} \frac{1}{x^{2}} \int_{0}^{x} f(t) \,d t$ is:

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

The equation $y^2e^{xy} = 9e^{-3}x^2$ defines $y$ as a differentiable function of $x$. The value of $\frac{dy}{dx}$ for $x = -1$ and $y = 3$ is

If $3 \sin xy + 4 \cos xy = 5$,then $\frac{dy}{dx}$ is equal to . . . . . .