Question Detail

This requires logarithmic differentiation.Take the natural logarithm of both sides:( \ln(y) = \ln(x^{\tan x}) )Using logarithm properties, bring the exponent down:( \ln(y) = \tan x \cdot \ln x )Now, differentiate both sides with respect to ( x ) using the chain rule on the left and the product rule on the right:( \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\tan x) \cdot \ln x + \tan x \cdot \frac{d}{dx}(\ln x) )( \frac{1}{y} \frac{dy}{dx} = (\sec^2 x) \ln x + \tan x \cdot \left(\frac{1}{x}\right) )Multiply both sides by ( y ):( \frac{dy}{dx} = y \left[ \sec^2 x \ln x + \frac{\tan x}{x} \right] )Substitute back ( y = x^{\tan x} ):( \frac{dy}{dx} = x^{\tan x} \left[ \frac{\tan x}{x} + \sec^2 x \ln x \right] )This matches option D.