Question Detail

To find the derivative of y = xtan x, we use logarithmic differentiation. Take the natural logarithm of both sides: ln y = ln(xtan x) = tan x * ln x.Now, 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)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)Rearranging the terms to match the options: dy/dx = xtan x * (tan x / x + sec2 x ln x).