Second derivative
where a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change.
is the second derivative of position (x) with respect to time.
The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:
When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written
The second derivative of a function f can be used to determine the concavity of the graph of f.[2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Similarly, a function whose second derivative is negative will be concave down (sometimes simply called concave), and its tangent line will lie above the graph of the function near the point of contact.
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa.
Specifically, The reason the second derivative produces these results can be seen by way of a real-world analogy.
Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration.
Clearly, the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse.
The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.
It is possible to write a single limit for the second derivative:
The limit is called the second symmetric derivative.
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
The limit above just gives a possibility for calculating the second derivative—but does not provide a definition.
The sign function is not continuous at zero, and therefore the second derivative for
Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are the same as those of f at a given point.
The formula for the best quadratic approximation to a function f around the point x = a is
This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.
For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the second derivative can be obtained.
and homogeneous Dirichlet boundary conditions (i.e.,
and the corresponding eigenvectors (also called eigenfunctions) are
For other well-known cases, see Eigenvalues and eigenvectors of the second derivative.
For a function f: R3 → R, these include the three second-order 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.
Another common generalization of the second derivative is the Laplacian.
The Laplacian of a function is equal to the divergence of the gradient, and the trace of the Hessian matrix.
