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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?

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    $\begingroup$ People are sometimes confused by the notation here. $e^{x^2}$ means $e^{(x^2)}$, not $\left(e^x\right)^2$. The latter would just be $e^{2x}$, but the former is not. $\endgroup$
    – lulu
    Commented Oct 16, 2021 at 13:43
  • $\begingroup$ Power towers are , by convention , calculated "from above" (or , if written in one line, "from right to left"). This is because the main purpose of power towers is creating huge numbers and with calculating them "from above" , we get the largest possible value. $\endgroup$
    – Peter
    Commented Oct 16, 2021 at 14:33
  • $\begingroup$ The rule you mean applies , if we have $(a^n)^m$ which is actually $a^{nm}$ , if $a$ is positive and $n,m$ real. $\endgroup$
    – Peter
    Commented Oct 16, 2021 at 14:35
  • $\begingroup$ @lulu thanks for clarifying so what is the difference between $(a^n)^m$ and $a^{n^m}$? I always thought that they are the same $\endgroup$
    – WilliamT
    Commented Oct 17, 2021 at 10:41
  • $\begingroup$ Just think through an example. Take $a=2,n=2,m=3$. Then $(a^n)^m=(2^2)^3=2^{2\times 3}=2^6=64$ but $a^{n^m}=a^{(n^m)}=2^{(2^3)}=2^8=256$. $\endgroup$
    – lulu
    Commented Oct 17, 2021 at 10:59

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There is no such power rule. What we have is $\left(a^n\right)^m=a^{nm}$. Besides, note that$$2^{2^3}=256\ne64=2^{2\times 3}.$$

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