higher-order derivative

If the first derivative is your speedometer (how fast position is changing), the second derivative is your accelerator (how fast the speed itself is changing). You can keep going: take the derivative of the derivative of the derivative, and so on. Each step asks 'how is the previous rate changing?' These are the higher-order derivatives.

Differentiating f gives f'; differentiating f' again gives the second derivative f'' (written f''(x), or d^2y/dx^2 in Leibniz notation); a third time gives f''', and beyond that people switch to f^(4), f^(5), and so on. In motion, if position is s(t), then s'(t) is velocity and s''(t) is acceleration. The notation d^2y/dx^2 is read 'd-squared y, d-x-squared' — a single symbol for 'differentiate twice', not an actual square.

The second derivative is the one you'll use most, because it reveals concavity — the way a curve bends. Where f'' > 0 the graph curves upward like a cup (concave up); where f'' < 0 it curves downward like a dome (concave down); and a point where the concavity switches is an inflection point. This is also why the second derivative test works: at a critical point where f' = 0, a positive f'' marks a local minimum (the cup holds water) and a negative f'' marks a local maximum.