Second derivatives are the second-order derivatives of a function with respect to its independent variable(s). In other words, they represent the rate of change of the first derivative or the acceleration of the original function. Second derivatives are essential in calculus for analyzing the curvature and concavity of functions, as well as in various applications, such as optimization and partial differential equations.
For a single-variable function, the second derivative is denoted as f”(x) or d²f/dx². For multivariable functions, second derivatives can be classified as pure second-order partial derivatives or mixed second-order partial derivatives, depending on whether the derivatives are taken with respect to the same or different independent variables.