Calculating Higher-Order Derivatives

Take a derivative and you get a new function. Nothing stops you from taking its derivative too. f'' means differentiate f, then differentiate the result; f''' means do it a third time. Each round strips degree from a polynomial term and eventually kills it, which is why a linear term like cx vanishes after two derivatives and a constant vanishes after one.

The physical reading makes this concrete. If position is s(t), velocity is s'(t), the rate position changes. Acceleration is s''(t), the rate velocity changes, which is why the second derivative measures bend: a curve can be climbing (positive first derivative) while curving downward into a cap shape (negative second derivative), because the slope itself is decreasing. Steepness lives in the first derivative; concavity lives in the second.

Products and trig functions make higher derivatives more work but not more mysterious. With e^kxsin(x), each differentiation needs the product rule again, and the exponential factor rides along unchanged in form while sin and cos keep trading places. With sin(x) alone, differentiating cycles through sin → cos → -sin → -cos → sin every four steps, so the hundredth derivative is predictable just by checking 100 mod 4. This cycling behavior is exactly what feeds into Taylor and Maclaurin series later, where every higher derivative at a point becomes a coefficient.