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$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$ $$=e^{e^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}\ln{x}} }$$ $$=e^{e^{ \ln{\left(\ln{x}\right)}} }$$ $$=e^{\ln{x} }$$ $$=x$$ $$f'(x)=1$$

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$$2(a^2 +b^2 ) = 2a^2 +2b^2$$ $$= a^2 + 2ab + b^2 + a^2 - 2ab + b^2$$ $$= (a+b)^2 +(a-b)^2$$

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$$\frac{n}{2}-\frac{n}{3}=30$$ $$\frac{n}{6}=30$$ $$n=180$$ $$\frac{n}{2}-\frac{n}{4}=30$$ $$\frac{n}{4}=30$$ $$n=120$$ $$\frac{n}{2}-\frac{n}{5}=30$$ $$\frac{3n}{10}=30$$ $$n=100$$ $$\frac{n}{3}-\frac{n}{4}=30$$ $$\frac{n}{12}=30$$ $$n=360$$ $$\frac{n}{3}-\frac{n}{5}=30$$ $$\frac{2n}{15}=30$$ $$n=225$$ $$\frac{n}{4}-\frac{n}{5}=30$$ $$\frac{n}{20}=30$$ $$n=600$$

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$$\sum_{i=1}^{2n}10^{i-1} a_i $$ $$\sum_{i=1}^{2n}10^{i-1} a_i + \sum_{i=1}^{2n}10^{2n-i} a_i = \sum_{i=1}^{2n}(10^{i-1}+10^{2n-i}) a_i $$ $$10^{i-1}+10^{2n-i} = 1+10^{2n-1} = \sum_{j=1}^{j=2n-2}9\times 10^j +11$$ $$=\sum_{j=1}^{j=n-1}99\times 10^{2j} +11$$ $$=11\left(\sum_{j=1}^{j=n-1}9\times 10^{2j} +1\right)$$

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