The second derivative measures how the slope of a function changes. For a quadratic function like f(x) = x², the first derivative gives us the slope at any point, while the second derivative tells us how that slope is changing.
The second derivative is mathematically defined as the derivative of the first derivative. We write it as f double prime of x, or d squared f over dx squared. For example, if f of x equals x cubed, then f prime of x equals 3x squared, and f double prime of x equals 6x.
Geometrically, the second derivative tells us about the concavity of a curve. When the second derivative is positive, the curve is concave up, like a smile. When it's negative, the curve is concave down, like a frown. When the second derivative equals zero, we may have an inflection point where the concavity changes.
The second derivative has important physical applications. In motion problems, if position is s of t, then velocity is the first derivative, and acceleration is the second derivative. For example, with position s equals 5t squared plus 2t plus 1, the acceleration is constant at 10 meters per second squared.
The second derivative test helps us classify critical points. When the first derivative equals zero at a point, we check the second derivative. If it's positive, we have a local minimum. If it's negative, we have a local maximum. For example, with f of x equals x cubed minus 3x, we find critical points at x equals plus or minus 1, giving us a maximum at negative 1 and a minimum at positive 1.