Real Analysis 44 | Higher Derivatives [dark version]

Higher derivatives are introduced as a structured way to measure how many times a function can be differentiated—and how smooth that differentiation remains—because that groundwork is essential for Taylor’s theorem later. Starting from a differentiable function f on an interval I, differentiating produces a new function f′. But f′ need not be differentiable again, and it may fail even to be continuous. When f′ exists and is continuous, f is called continuously differentiable; when f′ is itself differentiable, f becomes twice differentiable, and the process can be iterated to define higher derivatives f^{(n)}.

The transcript formalizes this using an inductive definition. It sets f^{(0)} = f, then defines “n times differentiable” by requiring that the (n−1)th derivative exists and is differentiable. From there, “n times continuously differentiable” means the nth derivative exists and is continuous. It also notes common alternative notations: Leibniz-style d^n f/dx^n and operator notation where differentiation acts on f. For infinite smoothness, “infinitely differentiable” is clarified as “arbitrarily often differentiable,” meaning f^{(n)} exists for every natural number n; in that case, all derivatives are continuous, so the function is continuously differentiable to any order.

To organize these smoothness classes, the transcript defines function spaces C^n(I) as the set of functions on I whose derivatives up to order n are continuous, and C^∞(I) as those differentiable to every order. These spaces form an inclusion chain: C(I) ⊇ C^1(I) ⊇ C^2(I) ⊇ … ⊇ C^∞(I). Concrete examples anchor the hierarchy: polynomials like f(x)=x^2 lie in C^∞(I) because higher derivatives eventually become the zero function, which still counts as “existing.” The exponential function is also highlighted as infinitely differentiable since its derivative reproduces the same functional form.

With higher derivatives in place, the transcript turns to a practical payoff: a sufficient condition for local extrema using the second derivative. For an interior point x0, a local maximum or minimum requires f′(x0)=0, but that condition alone is not sufficient. The missing ingredient is the existence of f′′(x0). If f′′(x0) > 0, then f has a local minimum at x0; if f′′(x0) < 0, then f has a local maximum. The proof for the “positive second derivative” case uses the definition of f′′(x0) as a limit (a difference quotient) to show that f′ changes sign from negative to positive around x0, implying f decreases on the left and increases on the right—exactly the behavior of a local minimum. The “negative second derivative” case follows by an analogous sign argument.

This combination—precise definitions of higher derivatives and a second-derivative extremum test—sets up the smoothness assumptions and derivative machinery needed for Taylor’s theorem in the next installment.