Calculating Higher-Order Derivatives

You already know f'(x) tells you the slope of f at each point, how fast the output is changing. But that slope function is itself a function, and it has its own rate of change. Take its derivative and you get f''(x), the second derivative, which measures how quickly the slope itself is changing. This is why acceleration is the derivative of velocity, which is the derivative of position: position tells you where you are, velocity tells you how fast that's changing, and acceleration tells you how fast the velocity is changing. There is nothing new to learn mechanically. You just apply the same differentiation rules again to whatever you got last time.

For polynomials this process eventually dies out completely. Every derivative lowers the degree of a polynomial by exactly one, because differentiating xⁿ gives n x^n-1. So a degree-5 polynomial becomes degree 4, then 3, then 2, then 1, then degree 0, a plain constant. One more derivative after that kills the constant too, since the derivative of any constant is 0, and all derivatives after that stay 0 forever. Trig and exponential functions behave differently: they often cycle. Differentiate sin(x) twice and you land back on a multiple of sin(x), never reaching zero, because sine and cosine just keep trading places under differentiation.

Higher-order derivatives connect directly to curve shape: the second derivative tells you concavity and is the tool behind the second derivative test and inflection points, which come right after this topic. The notation stacks the same way the operation does: f'(x), f''(x), f'''(x), then f⁴(x) and onward, or d^2y/dx² for the second derivative in Leibniz notation.