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)