Calculate the derivative of x^x

This problem is best solved using implicit differentiation and by tackling each side of the equation individually. First write down the problem as "y = x^x". Recalling that log rules can be used to simplify exponents we take the natural log of both sides and simplify to leave "ln(y) = xln(x)". The next step is to write the derivative term with respect to x on each side to leave "d(ln(y))/dx = d(xln(x))/dx". Applying the product rule to the RHS gives "d(ln(y))/dx = ln(x)dx/dx + xd(ln(x))/dx" this easily simplifies to "d(ln(y))/dx = ln(x) + x*(1/x)" or "d(ln(y))/dx = ln(x) + 1". Multiplying the LHS by dy/dy (equal to 1) gives "d(ln(y))/dy * dy/dx = ln(x)+1" which goes to "1/y * dy/dx = ln(x) + 1". Multiplying across the y gives "dy/dx = y(ln(x) + 1)". Now we can recall that y = x^x to complete the problem. leaving "dy/dx = x^x(ln(x) + 1)".