Question Detail

To differentiate a function of the form f(x)g(x), we use logarithmic differentiation.Let y = xtan x.Take the natural logarithm of both sides:ln(y) = ln(xtan x)Using the logarithm property ln(ab) = b ln(a):ln(y) = tan(x) ln(x)Now, differentiate both sides with respect to x. On the left, use the chain rule. On the right, use the product rule (fg)' = f'g + fg':(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)Now, 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 * [tan(x)/x + sec2(x) ln(x)]This matches option D.