I want to show some basic (suitable for Calculus I) but interesting characteristics of the derivative operator to my students; so far they have seen how to use the limit definition of the derivative to prove that the derivative of $\sin(x)$ is $\cos(x)$.
- I see that differentiation changes the evenness and oddness of functions
- Differentiation preserves the periodicity of functions, e.g. sines and cosines are $2\pi$ - periodic
- Starting with $\sin(x)$, if I differentiate it enough times w.r.t. $x$, I will get back the function $\sin(x)$. What can I say here?
What other interesting things can I say about the derivative operator? I feel my students are quickly getting bored of using the limit definition and are itching to use the "shortcuts", as they like to call it, i.e., they already know the formulas such as the power rule ... from their high school calculus course(s). Still, I have to stay on topic, according to what our department program coordinators want to be taught, so no "shortcuts" for them just yet ...