959 B
Executable file
959 B
Executable file
Exponential and logarithmic functions
The natural logarithm
The natural logarithm is defined as having its derivative equal to \frac{1}{x}. For x > 0, then
\frac{d}{dx} \ln x = \frac{1}{x}.
Standard limit
\lim_{h \to 0} \frac{\ln (1+h)}{h} = 1
The exponential function
The exponential function is defined as the inverse of the natural logarithm
\ln e^x = x.
Furthermore e may be defined by,
\begin{array}{ll}
\lim_{n \to \infty} (1 + \frac{1}{n})^n = e, \
\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x.
\end{array}
Derivative of exponential function
The derivative of y = e^x may be calculated by implicit differentation:
\begin{array}{ll}
y = e^x &\implies x = \ln y, \
&\implies 1 = \frac{1}{y} \frac{dy}{dx}, \
&\implies \frac{dy}{dx} = y = e^x.
\end{array}
Standard limit
\lim_{h \to 0} \frac{e^h - 1}{h} = 1