Given  y=cosx^(cosx^(cosx^...oo)), then evaluate dy/dx.

দেওয়া আছে, \(y = \cos x^{(\cos x^{(\cos x^{... \infty})})}\)

সুতরাং, \(y = (\cos x)^y\)

উভয় পক্ষে লগ নিয়ে পাই,

\(\ln y = \ln (\cos x)^y\)

\(\ln y = y \ln (\cos x)\)

এখন, x এর সাপেক্ষে অন্তরকলন করে পাই,

\(\frac{1}{y} \frac{dy}{dx} = y \cdot \frac{1}{\cos x} \cdot (-\sin x) + \ln (\cos x) \cdot \frac{dy}{dx}\)

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

\(\frac{dy}{dx} \left(\frac{1}{y} - \ln (\cos x)\right) = -y \tan x\)

\(\frac{dy}{dx} \left(\frac{1 - y \ln (\cos x)}{y}\right) = -y \tan x\)

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

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

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