Second Derivatives

Definition

Let $A \subset \R $ open. Let $f: A \to \R $ be differentiable with derivative $f': \R \to \R $. We call $f$ twice differentiable (or two times differentiable) if its derivative $f'$ is differentiable. In this case, we call the derivative of $f'$ the second derivative of $f$.

Notation

Let $A \subset R$. The second derivative of the twice-differentiable function $f: A \to \R $ is sometimes denoted $f''(x): A \to \R $