Use of the Difference Quotient
\[f(x)\]
from first principles. We define \[\frac{d(f(x))}{dx}= lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x} \]
.Example
\[ f(x)=x^n\]
.\[\begin{equation} \begin{aligned} \frac{d(f(x))}{dx} &= lim_{h \rightarrow 0} \frac{(x+h)^n-x^n}{(x+h)-x} \\ &= lim_{h \rightarrow 0} \frac{(x^n+nx^{n-1}h + \frac{n(n-1)}{2!}x^{n-2}h^2 + O(h^3))-x^n}{(x+h)-x} \\ & = nx^{n-1} \end{aligned} \end{equation}\]
.Analytic functions can be expressed as power series
\[\sum_n a_n x^n\]
- Mclaurin series. Because of this we can differentiate any analytic function using the difference quotient. 
