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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 ...

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  • $\begingroup$ It took a while for me to understand where all the parts took place, i.e. you choose two points in X and then you take their difference after mapping each to f(X) and then you rescale that difference using a scalar that depends on where they were/how far apart they were in X. $\endgroup$
    – Emil
    Oct 3 '17 at 5:24
  • $\begingroup$ @Emil, I feel they might already know this ... especially since I made them understand sort of rigorously what the sets domain and range were ... $\endgroup$
    – D.Hutchinson
    Oct 3 '17 at 6:02
  • $\begingroup$ I guess it sometimes could be how long the differences are relative to eachother times a normalized direction too... Don't know if that's cool. $\endgroup$
    – Emil
    Oct 3 '17 at 6:50
  • $\begingroup$ Have you told them that it is linear? Another interesting thing is that not every function is the derivative of a simple function $\frac{\sin x}{x}$ for instance. BTW doing derivative without rules, just applying the limit, is like driving a car only when you have built it from the wheels... $\endgroup$
    – Raffaele
    Oct 3 '17 at 6:52
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The most important: the derivative gives the locally best linear approximation: $$f(x) \approx f'(c)(x-c) + f(c).$$ Also interesting: a geometric interpretation of properties deduced from the $\lim$ definition. Your two first examples are ideal for this.

About the $\sin$ example: use as illustration of differential equations ($\sin$ is one solution of $y'' = y$). Without deepening, tell them about the importance of differential equations in physics.

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