E1:

Try to understand what the higher derivatives mean, as applied to the base curve of a function

E2:  Do the same for the sine function. Why does the 4th derivative look like the base function? (Hint:  reason first why the second derivative looks like the negative of the base function, and then apply the same reasoning to the fourth and second derivative).

E3:  Go through the derivatives of the power function:  What type of parabola is each one?

(Hint:  the easy way is to reason downward from the 9th one).

E4:  Choose sine(x2). Consider why the derivatives assume higher and higher values (observe that the y scale is self adjusting!). Why are discernable oscillations shifting to higher x values with increasing order?

E5:  With sine(x)/x many derivatives are within comparable amplitude range, while looking quite different. Go through exercise E1 to understand the change in the basic function expressed by the different derivatives.

(Hint:  start by comparing any two consecutive derivatives)