Question Detail

This requires logarithmic differentiation:Given y = xtan xTake the natural logarithm of both sides: ln(y) = ln(xtan x)Using logarithm properties: ln(y) = tan x * ln xNow, differentiate both sides with respect to x. Use the chain rule 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 = sec2x * ln x + tan x * (1/x)Multiply both sides by y:dy/dx = y * (sec2x * ln x + tan x / x)Substitute y = xtan x back:dy/dx = xtan x * (sec2x * ln x + tan x / x)This matches option D.