Week 3, Short Day: Differentiability

In highschool you have probably seen the derivative introduced as the limit of a difference quotient: $\(f'(x_0)=\frac{f(x)-f(x_0)}{x-x_0}\,\,\,\mathrm{for}\,\,\,x\rightarrow x_0\,.\)$ Today we will as an alternative approach introduce differentiability using epsilon functions, which will be needed later on for generalizing differentiability to functions of multiple variables. As a prelude we will differentiate complex functions of a single real variable. By doing this we become able to construct the derivative of an important case of the complex exponential function. Behind it all lie the usual computational rules for the derivative. It is crucial that you master these, and that you understand how differentiation in general is performed!