Question Detail

This is a function of the form f(x)g(x), so we use logarithmic differentiation.Let y = xtan x.Take the natural logarithm of both sides: ln y = tan x * ln x.Now, differentiate both sides with respect to x, using implicit differentiation on the left and the product rule on the right:(1/y) dy/dx = (d/dx tan x) * ln x + tan x * (d/dx ln x)(1/y) dy/dx = (sec² x) * ln x + tan x * (1/x)Multiply both sides by y:dy/dx = y * (sec² x ln x + tan x / x)Substitute y = xtan x back into the equation:dy/dx = xtan x (sec² x ln x + tan x / x)This matches option D.