Differentiation of Infinite Series

If $y=\sqrt{x+\sqrt{x+\sqrt{x+...\mathrm{to}\infty }}}$,  prove that $\frac{dy}{dx}=\frac{1}{2y-1}$?

$y={\left(\sin x\right)}^{{\left(\sin x\right)}^{{\left(\sin x\right)}^{...\infty }}}$, prove that  $\frac{y^2\cot x}{\left(1-y \mathrm{logsin}x\right)}$ 

If $y={\left(\tan x\right)}^{{\left(\tan x\right)}^{{\left(\tan x\right)}^{...\infty }}}$, prove that $\frac{dy}{dx}=2$ at $x=\frac{\pi }{4}$ 

If $y={\left(\cos x\right)}^{{\left(\cos x\right)}^{\left(\cos x\right)...\infty }}$, prove that $\frac{dy}{dx}=-\frac{y^2\tan x}{\left(1-y \mathrm{logcos}x\right)}$?

If $y=e^{x^{e^x}}+x^{e^{e^x}}+e^{x^{x^e}}$, prove that $\frac{dy}{dx}=e^{x^{e^x}}·x^{e^x}\left\{\frac{e^x}{x}+e^x·\log x\right\}+x^{e^{e^x}}·e^{e^x}\left\{\frac{1}{x}+e^x·\log x\right\}+e^{x^{x^e}}{x}^{x^e}·x^{e-1}\left\{x+e\log x\right\}$

If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+...to \infty }}}$, prove that $\left(2y-1\right)\frac{dy}{dx}=\frac{1}{x}$ 

If $y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x+..to \infty }}}$ , prove that $\frac{dy}{dx}=\frac{{\sec }^{2}x}{2y-1}$

If $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+...to \infty }}}$ , prove that $\frac{dy}{dx}=\frac{\sin x}{1-2y}$