Question

If $y=(\cos x{)}^{(\cos x{)}^{(\cos x{)}^{\cdots \infty }}}$, then find the value of $\frac{dy}{dx}$ at $x=\frac{\pi }{2}$

Answer 1

$y=(\cos x{)}^{(\cos x{)}^{(\cos x{)}^{\cdots \infty }}}$

$\therefore y=(\cos x{)}^y$

Apply on both sides

$\ln y=y\ln \cos x$

$\frac{1}{y}\frac{dy}{dx}=\frac{dy}{dx}\ln \cos x-y\tan x$

$(1-y\ln \cos x)\frac{dy}{dx}=-y^2\tan x$

$\frac{dy}{dx}=\frac{-y^2\tan x}{1-\ln y}$ $(\because y\ln \cos x=\ln y)$

At $x=\frac{\pi }{2},\frac{dy}{dx}$ is not defined, as $\left(\ln y\right)$ and $\tan x\to \infty$ as $x\to \frac{\pi }{2}$