the distributional derivative
Classical differentiation refuses to act on rough things — a corner has no slope, a jump has no derivative, and a point source is not even a function. The distributional derivative is a way of differentiating that never refuses. It assigns a derivative to every distribution, and crucially it agrees with the ordinary derivative whenever the ordinary one exists. So it loses nothing and extends differentiation to objects that classically had none.
The definition is a single inspired move: shift the derivative off the rough object and onto the smooth probe. For ordinary functions, integration by parts says the integral of u' times phi equals minus the integral of u times phi', because phi vanishes at the ends (it has compact support). Take that identity as the definition: the derivative of a distribution T is the distribution T' whose reading against any probe is (T', phi) = -(T, phi'). The probe is smooth, so phi' always exists; therefore T' always exists. Differentiating again just moves another derivative onto phi, so every distribution can be differentiated infinitely many times, with no loss of regularity ever — a startling contrast with classical analysis.
This is the engine of the whole subject. The derivative of the Heaviside step is the delta; the derivative of a function with a corner is its left-and-right slopes plus nothing at the corner; the second derivative of the absolute value is twice the delta. Weak solutions of PDEs are defined by letting derivatives act in this distributional sense, which is exactly what makes shocks, point sources, and non-smooth data admissible.
Take u(x) = |x|. Classically its derivative is -1 for x < 0 and +1 for x > 0 with a corner at 0. As a distribution that derivative is the sign function, and differentiating once more gives u'' = 2 delta: the corner in |x| shows up as a delta of strength 2 exactly at the kink.
Every kink and jump is recorded, not erased — the distributional derivative measures it as a delta.
If a function has an honest classical derivative everywhere, its distributional derivative is just that — no surprises. The extra terms (deltas) appear only at jumps and corners, where the classical derivative was undefined or missed the jump.