BÀI 4: Hàm số mũ, hàm số logarit

3) Giới hạn đặc biệt

\(\mathop {\lim }\limits_{x \to 0} {(1 + x)^{\frac{1}{x}}} = \mathop {\lim }\limits_{x \to \pm \infty } {\left( {1 + \frac{1}{x}} \right)^x} = e\) · \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln (1 + x)}}{x} = 1\) · \(\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - 1}}{x} = 1\)

4) Đạo hàm

\({\left( {{a^x}} \right)^\prime } = {a^x}\ln a\); \({\left( {{a^u}} \right)^\prime } = {a^u}\ln a.u\prime \)
\({\left( {{e^x}} \right)^\prime } = {e^x}\); \({\left( {{e^u}} \right)^\prime } = {e^u}.u\prime \)
\({\left( {{{\log }_a}\left| x \right|} \right)^\prime } = \frac{1}{{x\ln a}}\); \({\left( {{{\log }_a}\left| u \right|} \right)^\prime } = \frac{{u\prime }}{{u\ln a}}\)
\({\left( {\ln \left| x \right|} \right)^\prime } = \frac{1}{x}\) (x > 0); \({\left( {\ln \left| u \right|} \right)^\prime } = \frac{{u\prime }}{u}\)