For the question e^(xlnx), the derivative is given as x^x(1+lnx). I understand where the (1+lnx) comes from, that is the derivative of xlnx, but I'm not sure why it's being multiplied by x^x. I thought that d/dx(e^u) = u' e^u

Second, for the question y=lnx at the point where x=e/2, find the equation of the tangent at that point. Isn't the derivative and slope here 1/2e? The equation given as the answer is y = (2/e)x