In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation: where a is acceleration, v is velocity, t is time, x is position, and d is the instantanteous "delta" or change. The last expression is the second derivative of position with respect to time. On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. 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.
Given the function the derivative of f is the function The second derivative of f is the derivative of f′, namely
Relation to the graph
Concavity
The second derivative of a function f measures the concavity of the graph of f. A function whose second derivative is positive will be concave up, meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down, and its tangent lines will lie above the graph of the function.
Inflection points
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. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.
The relation between the second derivative and the graph can be used to test whether a stationary point for a function is a local maximum or a local minimum. Specifically,
If then has a local maximum at.
If then has a local minimum at.
If, the second derivative test says nothing about the point, a possible inflection point.
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.
Limit
It is possible to write a single limit for the second derivative: The limit is called the second symmetric derivative. Note that the second symmetric derivative may exist even when the second derivative does not. The expression on the right 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. Please note that 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. As a counterexample look on the sign function which is defined through 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 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.
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.
