Answer
$${x^{{x^{10}} + 9}}\left( {1 + 10\ln x} \right)$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dx}}\left( {{x^{\left( {{x^{10}}} \right)}}} \right) \cr
& {\text{we use the inverse relationship }}{e^{\ln x}} = x \cr
& = {e^{\ln {x^{\left( {{x^{10}}} \right)}}}} \cr
& = {e^{{x^{10}}\ln x}} \cr
& {\text{evaluate the derivative}} \cr
& = \frac{d}{{dx}}\left( {{e^{{x^{10}}\ln x}}} \right) \cr
& {\text{by }}\frac{d}{{dx}}\left( {{e^{u\left( x \right)}}} \right) = {e^{u\left( x \right)}}u'\left( x \right) \cr
& = {e^{{x^{10}}\ln x}}\frac{d}{{dx}}\left( {{x^{10}}\ln x} \right) \cr
& {\text{product rule}} \cr
& = {e^{{x^{10}}\ln x}}\left( {\frac{{{x^{10}}}}{x} + 10{x^9}\ln x} \right) \cr
& {\text{simplify}} \cr
& = {x^{\left( {{x^{10}}} \right)}}\left( {{x^9} + 10{x^9}\ln x} \right) \cr
& = {x^{{x^{10}} + 9}}\left( {1 + 10\ln x} \right) \cr} $$
