Differentiate x^x
Write y = x^x , then we need to find dy/dx. Take the natural logarithm of both sides to give ln(y) = ln (x^x). Using log laws we can write the RHS as x ln (x). Now differentiate both sides with respect to x. The left hand side needs implicit differentiation so differentiate ln(y) with respect to y and then multiply it by dy/dx so the LHS is now 1/y * dy/dx . Use the product rule on the RHS to get ln x +1 on the RHS. now we have 1/y *dy/dx = ln(x) +1 . Multiply through by y to get dy/dx = y(ln(x) +1) then use the fact that y = x^x to finally write that dy/dx = (ln(x) +1)*x^x
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