Second derivative

The derivative of a function’s first derivative, measuring how the function’s rate of change varies. Its sign indicates local concavity where the derivative exists.

First derivative: The derivative of a function, giving its instantaneous rate of change where it exists. The second derivative is obtained by differentiating this function again.
Differentiation: The process of finding a function’s derivative. A second derivative applies differentiation twice in succession.
Acceleration: The rate of change of velocity with respect to time. For position as a function of time, acceleration is the second derivative.
First derivative test: A method for classifying stationary points by examining sign changes in the first derivative. Unlike the second derivative test, it can classify extrema when the second derivative test is inconclusive.
Concavity: A function’s tendency to bend upward or downward over an interval. The sign of the second derivative identifies concavity on intervals where it is continuous.
Leibniz notation: A notation for derivatives that expresses changes in dependent and independent variables. The symbol d²y/dx² denotes the second derivative with respect to x.
Optimization: The study of finding inputs that maximize or minimize an objective function. Second derivatives help classify stationary points and assess local curvature.
Central difference: A numerical approximation to a derivative based on function values on both sides of a point. Finite differences estimate curvature from samples when analytic second derivatives are unavailable.
Inflection point: A point where a function changes from concave up to concave down, or vice versa. A sign change in the second derivative can reveal an inflection point.
Higher-order derivative: A derivative obtained by differentiating a function repeatedly. The second derivative is the first higher-order derivative beyond the ordinary derivative.

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