Question Detail

To differentiate y = xtanx, we use logarithmic differentiation.Take the natural logarithm of both sides: ln(y) = ln(xtanx) = tan(x) ∙ ln(x).Now, differentiate both sides with respect to x using the product rule on the right side:d/dx [ln(y)] = d/dx [tan(x) ∙ ln(x)](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 back y = xtanx:dy/dx = xtanx ∙ (sec2(x) ∙ ln(x) + tan(x)/x)This matches option D.