Question Detail

This problem requires logarithmic differentiation because the variable x appears in both the base and the exponent.1. Take the natural logarithm of both sides: ln y = ln(xtan x) = tan x * ln x.2. Differentiate both sides with respect to x. Use 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)3. Multiply by y to solve for dy/dx:dy/dx = y * ( (sec2 x) ln x + tan x / x )4. Substitute y = xtan x back into the equation:dy/dx = xtan x * ( (sec2 x) ln x + tan x / x )This matches option D.