I'm calculating the 3rd degree Taylor polynomial for $y =$ $x^{x^x}$ in $x_0=1$. I calculated the first derivative, which is $e^{lnx*x^x}*(\frac{x^x}{x}+lnx*e^{lnx*x}*(1+lnx))$. Finding the first degree derivative in $x_0=1$ is very easy, but calculating the second and third derivative is not.
Is there any way I can simplify it or do I really have to find all the lengthy second and third degree derivatives of $y$?
Thanks