User:Michael Hardy/compactness

${\begin{aligned}&{\frac {d}{dx}}e^{x}={\big (}e^{x}\cdot 1{\big )}={\big (}e^{x}\times {\text{constant}}{\big )}=e^{x}\cdot \ln e\\[12pt]&{\frac {d}{dx}}2^{x}\approx {\big (}2^{x}\times 0.69{\big )}={\big (}2^{x}\times {\text{constant}}{\big )}=2^{x}\cdot \ln 2\\[12pt]&{\frac {d}{dx}}10^{x}\approx {\big (}10^{x}\times 2.3{\big )}={\big (}10^{x}\times {\text{constant}}{\big )}=10^{x}\cdot \ln 10\\[12pt]&{\frac {d}{dx}}10^{x}={\frac {d}{dx}}{\big (}e^{\ln 10}{\big )}^{x}={\frac {d}{dx}}e^{(\ln 10)x}=e^{(\ln 10)x}\cdot {\frac {d}{dx}}{\big (}(\ln 10)x{\big )}=e^{(\ln 10)x}\cdot \ln 10=10^{x}\cdot \ln 10\end{aligned}}$