can be written as a difference quotient of difference quotients: This limit can be viewed as a continuous version of the second difference for sequences. However, the existence of the above limit does not mean that the function has a second derivative. The limit above just gives a possibility for calculating the second derivative—but does not provide a definition. A counterexample is the sign function , which is defined as: The sign function is not continuous at zero, and therefore the second derivative for does not exist. But the above limit exists for : Quadratic approximation Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function . This is the quadratic function whose first and second derivatives are the same as those of at a given point. The formula for the best quadratic approximation to a function around the point is This quadratic approximation is the second-order Taylor polynomial for the function centered at . Eigenvalues and eigenvectors of the second derivative For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the second derivative can be obtained. For example, assuming and homogeneous Dirichlet boundary conditions (i.e., ), the eigenvalues are and the corresponding eigenvectors (also called eigenfunctions) are . Here, For other well-known cases, see Eigenvalues and eigenvectors of the second derivative. Generalization to higher dimensions The Hessian The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function f: R3 → R, these include the three second-order partials and the mixed partials If the function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test.) The Laplacian Another common generalization of the second derivative is the Laplacian. This is the differential operator (or ) defined by The Laplacian of a function is equal to the divergence of the gradient, and the trace of the Hessian matrix. See also Chirpyness, second derivative of instantaneous phase Finite difference, used to approximate second derivative Second partial derivative test Symmetry of second derivatives References Further reading Print Online books External links Discrete Second Derivative from Unevenly Spaced Points Mathematical analysis Differential calculus Functions and mappings Linear