Question Detail

To differentiate y = xtanx, use logarithmic differentiation.Take the natural logarithm of both sides: ln y = ln(xtanx) = tanx * ln x.Differentiate both sides with respect to x using the product rule on the right side:(1/y) dy/dx = (d/dx(tanx)) * ln x + tanx * (d/dx(ln x))(1/y) dy/dx = sec2x * ln x + tanx * (1/x).Now, solve for dy/dx:dy/dx = y * (sec2x * ln x + (tanx)/x).Substitute y = xtanx back into the equation:dy/dx = xtanx ( sec2x * ln x + (tanx)/x ).This matches option D.