PART 14：例題-指數的指數

設 \(f(x) = {2^{{2^x}}}\) ，求 \(f'(x)\)

SOL:

\(f(x) = {2^{{2^x}}}\)，

\(\ln f(x) = \ln {2^{{2^x}}}\) ，

\(\ln f(x) = {2^x}\ln 2\) ，等號兩邊微分

\(\frac{1}{{f(x)}}f'(x) = {2^x}(\ln 2)(\ln 2)\) ，

\(f'(x) = f(x){2^x}(\ln 2)(\ln 2) = {2^{{2^x}}}{2^x}{(\ln 2)^2}\)