exponential function
general exponential function
$${f(x) = a^x}$$
For what a is the slope of the tangent equal to 1 ?
trying to differentiate the general exponential function
applying the differential quotient:
$${\begin{align} f’(x) & = \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h} \newline & = \lim\limits_{h \to 0} \frac{a^{x+h} - a^x}{h} \newline & = \lim\limits_{h \to 0} a^x \cdot \frac{a^{h} - 1}{h} \newline & = a^x \cdot \lim\limits_{h \to 0} \frac{a^{h} - 1}{h} \end{align}}$$
it can be shown that $${\lim\limits_{h \to 0} \frac{a^{h} - 1}{h} = \ln{a}}$$
which we call the natural logarithm of a.
Logarithms have an older history than Euler’s number e. Napier’s tables of Logarithms are one of the most important advances in this area.
Also, early work on logarithms links the area beneath the graph of the function $${f(x) = \frac{1}{x}}$$
see the quadrature of the hyperbola
Euler’s number e
If we set the limit in above derivative equal to 1 we have following definition $${\lim_{h \to 0} {\frac{e^h - 1}{h}} = 1}$$ where e is the base when this is true.
The number e can also be obtained if you investigate 100% compound interest given at every instant.
$${e = \lim_{n \to \infty} {\left( 1 + \frac{1}{n} \right)^n} \approx 2.718281828459…}$$
graph of the e-function
derivative of ex
as e is defined as
$${\lim_{h \to 0} {\frac{e^h - 1}{h}} = 1}$$
the derivative of the exponential function $f(x) = e^x$ is
$${f’(x)=e^x \cdot \lim_{h \to 0} {\frac{e^h - 1}{h}} = e^x \cdot 1 = e^x}$$
examples
alternative definitions
$${e = \sum_{n = 0}^{\infty} {\frac{1}{n!}} = {\frac{1}{0!}} + {\frac{1}{1!}} + {\frac{1}{2!}} + {\frac{1}{3!}} + …}$$