Question

Solve if (tan y)^x = (sec x)^y , find dy/dx

Answer 1

Given,

$(\tan y)^{x}=(\sec x)^{y}\;\;\\ \\\text{taking log}\;\\ \\x\log\tan y=ylog\sec x\;\\\;\\\text{Diff. both sides  w.r.t, x}\;\\\;\\\;x\frac{d\log\tan y}{dx}+\log\tan y=y\frac{d\log\sec x}{dx}+\log\sec x\frac{dy}{dx}\\\;\\\;x\frac{1}{\tan y}\frac{d\tan y}{dx}=y\frac{1}{\sec x}\sec x.\tan x}+\log\sec x\frac{dy}{dx}\\\;\\\;\frac{x}{\tan y}\sec^{2}y.\frac{dy}{dx}=y\tan x+\log\sec x\frac{dy}{dx}\\\;\\(\frac{x\sec^{2}y}{\tan y}-\log\sec x)\frac{dy}{dx}=y\tan x\\\;\\;\frac{dy}{dx}=\frac{y\tan x}{\frac{x\sec^{2}y}{\tan y}-\log\sec x}\\\;\\\frac{dy}{dx}=\frac{y.\tan y.\tan x}{{x\sec^{2}y}-\tan y\log\sec x}$