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 = ln(xtan x)ln y = tan x * ln xNow, differentiate implicitly with respect to x using the product rule on the right side:(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)Now, solve for dy/dx by multiplying both sides by y:dy/dx = y * (sec2 x ln x + tan x / x)Substitute back y = xtan x:dy/dx = xtan x (sec2 x ln x + tan x / x)Rearranging the terms in the parenthesis to match the options, we get:dy/dx = xtan x (tan x / x + sec2 x ln x)Therefore, the correct option is D.