higher-order derivative
If the first derivative is your speed, the second derivative is your acceleration — the rate at which the speed itself is changing. Differentiation can be iterated: take the derivative of the derivative, then of that, and so on, each pass revealing a finer layer of how a quantity bends and curves.
Definition: the second derivative is f''(x) = (f')'(x), the derivative of f' wherever that exists; inductively, the n-th derivative f^(n) is the derivative of f^(n-1). For f^(n)(a) to be defined, f^(n-1) must exist in a neighborhood of a (not merely at a) and be differentiable at a. Notations include f''(a), f^(n)(a), and the Leibniz forms d^2f/dx^2 and d^n f/dx^n.
Higher derivatives interpret the geometry beyond slope: the sign of f'' governs concavity (concave up where f'' > 0, concave down where f'' < 0) and underlies the second-derivative test for extrema. A subtle point: each order demands more regularity than the last, and existence at one level does not guarantee niceness at the next — f^(n) can exist while f^(n+1) does not, or while f^(n) itself is discontinuous.
For f(x) = x^4 we get f'(x) = 4x^3, f''(x) = 12x^2, f'''(x) = 24x, f^(4)(x) = 24, and f^(5)(x) = 0 thereafter.
A polynomial of degree n has all derivatives, with the (n+1)-th and beyond identically zero.
A function with continuous derivatives up to order k is called C^k. The chain C^0 (continuous) ⊇ C^1 ⊇ C^2 ⊇ ... is strict: each containment can be witnessed by an explicit function in one class but not the next.