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Imagine riding a roller coaster with your eyes closed: even if your altitude is not changing, you instantly feel whether the track is curving up under your feet or dropping away beneath you.
While the first derivative tells you the slope of a curve, the second derivative measures how that slope itself is changing from point to point.

The Second Derivative

The second derivative is simply the derivative of the first derivative, written as f′′(x)f''(x) or d2ydx2\frac{d^2y}{dx^2}.
In the 1680s, Gottfried Wilhelm Leibniz developed the notation d2ydx2\frac{d^2y}{dx^2} to represent taking the rate of change of dydx\frac{dy}{dx} across an infinitesimal step in xx.
How does the value of this second derivative shape what the graph looks like?

Concavity: Bending Up vs Bending Down

Concavity describes the direction in which a curve bends. When tangent slopes increase from left to right, the curve bends upward like a bowl.
A graph is concave up where f′′(x)>0f''(x) > 0 (tangents lie below the curve), and concave down where f′′(x)<0f''(x) < 0 (tangents lie above the curve).
What happens at the exact point where a curve transitions from bending downward to bending upward?

Points of Inflection