The derivative of y = xln(x) with respect to x is dy/dx = ln(x) + 1. This result can be obtained by using the product rule and the well-known results d(ln(x))/dx = 1/x and dx/dx = 1.
The product rule of differentiation states that the derivative of a product of two functions f(x) and g(x) is given by f(x)g'(x) + g(x)f'(x), where f'(x) is the derivative of f(x) with respect to x and g'(x) is the derivative of g(x) with respect to x. Letting f(x) = x and g(x) = ln(x), the mathematician can derive f'(x) = 1 and g'(x) = 1/x. Substituting these values into the formula gives d(xln(x))/dx = ln(x) + 1.