Multivariate calculus from first principles
Just as a 1-D function has a second derivative, a multivariable function has second-order partials. You differentiate twice. The new wrinkle is that you now get to choose which variable to differentiate by each time, and something tidy happens when you mix them.
The pure second partials ∂²f/∂x² and ∂²f/∂y² measure curvature along each axis. The mixed partial ∂²f/∂x∂y differentiates first by y, then by x; it measures how the slope in one direction changes as you move in the other.
A first partial tells you the hillside's steepness; a second partial tells you how that steepness itself is changing as you move, which is the slope's curvature. Walking east, does the ground keep getting steeper or start to level off? That bending of the eastward slope ∂f/∂x as you press further east is the second partial ∂²f/∂x², the curvature of the hill along that direction.