By the power rule for exponents $a^{n^m}=a^{nm}$ so for example $2^{5^2}= 2^{10}$.
However, when we try to calculate $d/dx [ e^{x^2}]$ the correct answer is $2xe^{x^2}$ which confuses me because by the power rule I would think that we can rewrite $d/dx [ e^{x^2}]$ as $d/dx [ e^{2x}]$ for which the result would be $2e^{2x}$. Can someone please clarify why this does not work when it comes to derivation?
There is no such power rule. What we have is $\left(a^n\right)^m=a^{nm}$. Besides, note that$$2^{2^3}=256\ne64=2^{2\times 3}.$$