Single-variable calculus from first principles
If the first derivative f′ tells you the slope, what does the derivative of the slope tell you? That's the second derivative f″, and it measures how the slope is changing, which is the curve's concavity.
Just differentiate twice. For f(x) = x³: first f′ = 3x², then f″ = 6x. You can keep going (third, fourth derivatives) each one differentiating the last.
The sign of f″ tells you which way the curve bends. If f″ > 0 the curve is concave up: it cups upward like a bowl (∪), and the slope is increasing. If f″ < 0 it's concave down: it caps over like a dome (∩), and the slope is decreasing. Where the concavity flips is an inflection point.