Question Detail

To differentiate a function of the form f(x)g(x), we use logarithmic differentiation. Take the natural logarithm of both sides: ln y = ln(xtan x). Using the logarithm property ln(ab) = b ln a, we get ln y = tan x * ln x. Now, differentiate both sides with respect to x. On the left, d⁄dx(ln y) = (1⁄y) dy⁄dx. On the right, use the product rule: d⁄dx(tan x * ln x) = (d⁄dx(tan x)) * ln x + tan x * (d⁄dx(ln x)). This gives (sec2 x) * ln x + tan x * (1⁄x). So, (1⁄y) dy⁄dx = sec2 x ln x + (tan x)⁄x. Multiply both sides by y, and substitute y = xtan x back: dy⁄dx = xtan x (sec2 x ln x + (tan x)⁄x). Rearranging the terms inside the parenthesis gives option D.