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{{Calculus |Differential}}
[[File:4 fonctions du second degré.svg|right|thumb|200px|The second derivative of a [[quadratic function]] is [[constant function|constant]].]]

In [[calculus]], the '''second derivative''', or the '''second order derivative''', of a [[function (mathematics)|function]] ''f'' is the [[derivative]] of the derivative of ''f''. 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 a vehicle with respect to time is the instantaneous [[acceleration]] of the vehicle, or the rate at which the [[velocity]] of the vehicle is changing.

On the [[graph of a function]], the second derivative corresponds to the [[curvature]] or concavity of the graph. The graph of a function with positive second derivative curves upwards, while the graph of a function with negative second derivative curves downwards.

== Second derivative power rule ==
The power rule for the first derivative, if solved down a bit, will produce the second derivative power rule. The rule is given below:

<math>\frac{d^2}{dx^2}[x^n]=n(n-1)x^{(n-2)}=(n^2-n)x^{(n-2)}.</math>

== Notation ==
{{Details|Notation for differentiation}}
The second derivative of a function <math>f(x)\!</math> is usually denoted <math>f''(x)\!</math>. That is:
:<math>f'' = (f')'\!</math>
When using [[Leibniz's notation]] for derivatives, the second derivative of a dependent variable ''y'' with respect to an independent variable ''x'' is written
:<math>\frac{d^2y}{dx^2}.</math>
This notation is derived from the following formula:
:<math>\frac{d^2y}{dx^2} \,=\, \frac{d}{dx}\left(\frac{dy}{dx}\right).</math>

== Example ==
Given the function
:<math>f(x) = x^3,\!</math>
the derivative of ''f'' is the function
:<math>f'(x) = 3x^2.\!</math>
The second derivative of ''f'' is the derivative of ''f''′, namely
:<math>f''(x) = 6x.\!</math>

== Relation to the graph ==
[[File:Animated illustration of inflection point.gif|500px|thumb|A plot of <math>f(x) = \sin(2x)</math> from <math>-\pi/4</math> to <math>5\pi/4</math>. The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0, <math>\pi</math>/2, and <math>\pi</math>).]]

=== 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]] (sometimes referred to as convex), 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]] (sometimes called simply “concave”), and its tangent lines will lie above the graph of the function.

=== Inflection points ===
{{main|Inflection point}}
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.

=== Second derivative test ===
{{main|Second derivative test}}
The relation between the second derivative and the graph can be used to test whether a [[stationary point]] for a function (i.e. a point where <math>f'(x)=0\!</math>) is a [[local maximum]] or a [[local minimum]]. Specifically,
* If <math>\ f^{\prime\prime}(x) < 0</math> then <math>\ f</math> has a local maximum at <math>\ x</math>.
* If <math>\ f^{\prime\prime}(x) > 0</math> then <math>\ f</math> has a local minimum at <math>\ x</math>.
* If <math>\ f^{\prime\prime}(x) = 0</math>, the second derivative test says nothing about the point <math>\ x</math>, 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 (mathematics)|limit]] for the second derivative:
:<math>f''(x) = \lim_{h \to 0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.</math>
The expression on the right can be written as a [[difference quotient]] of difference quotients:
:<math>\frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = \frac{\frac{f(x+h) - f(x)}{h} - \frac{f(x) - f(x-h)}{h}}{h}.</math>
This limit can be viewed as a continuous version of the [[second difference]] for [[sequence (mathematics)|sequences]].

Please note that the existence of the above limit does not mean that the function <math>f</math> 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]] <math>\sgn(x)</math> which is defined through

:<math>\sgn(x) = \begin{cases}
-1 & \text{if } x < 0, \\
0 & \text{if } x = 0, \\
1 & \text{if } x > 0. \end{cases}</math>

The sign function is not continuous at zero and therefore the second derivative for <math>x=0</math> does not exist. But the above limit exists for <math>x=0</math>:

:<math>\begin{align}
\lim_{h \to 0} \frac{\sgn(0+h) - 2\sgn(0) + \sgn(0-h)}{h^2} &= \lim_{h \to 0} \frac{1 - 2\cdot 0 + (-1)}{h^2} \\
&= \lim_{h \to 0} \frac{0}{h^2} \\
&= 0 \end{align}</math>

== Quadratic approximation ==
Just as the first derivative is related to [[linear approximation]]s, 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
:<math>f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2}f''(a)(x-a)^2.</math>
This quadratic approximation is the second-order [[Taylor polynomial]] for the function centered at ''x'' = ''a''.

== 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 <math>x \in [0,L]</math> and homogeneous [[Dirichlet boundary conditions]], i.e., <math> v(0)=v(L)=0</math>, the [[eigenvalues]] are <math> \lambda_j = -\frac{j^2 \pi^2}{L^2}</math> and the corresponding [[eigenvectors]] (also called [[eigenfunctions]]) are <math> v_j(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{j \pi x}{L}\right) </math>. Here, <math> v''_j(x) = \lambda_j v_j(x), \, j=1,\ldots,\infty.</math>

For other well-known cases, see the main article [[eigenvalues and eigenvectors of the second derivative]].

== Generalization to higher dimensions ==

=== The Hessian ===
{{main|Hessian matrix}}
The second derivative generalizes to higher dimensions through the notion of second [[partial derivative]]s. For a function ''f'':'''R'''<sup>3</sup> → '''R''', these include the three second-order partials

:<math>\frac{\part^2 f}{\part x^2}, \; \frac{\part^2 f}{\part y^2}, \text{ and }\frac{\part^2 f}{\part z^2}</math>

and the mixed partials

:<math>\frac{\part^2 f}{\part x \, \part y}, \; \frac{\part^2 f}{\part x \, \part z}, \text{ and }\frac{\part^2 f}{\part y \, \part z}.</math>

If the function's image and domain both have a potential, then these fit together into a [[symmetric matrix]] known as the '''Hessian'''. The [[eigenvalue]]s 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 ===
{{main|Laplace operator}}
Another common generalization of the second derivative is the '''Laplacian'''. This is the differential operator <math>\nabla^2</math> defined by
:<math>\nabla^2 f = \frac{\part^2 f}{\part x^2}+\frac{\part^2 f}{\part y^2}+\frac{\part^2 f}{\part z^2}.</math>
The Laplacian of a function is equal to the [[divergence]] of the [[gradient]].

== References ==
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=== Online books ===

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== External links ==
*[http://mathformeremortals.wordpress.com/2013/01/12/a-numerical-second-derivative-from-three-points/ Discrete Second Derivative from Unevenly Spaced Points]

[[Category:Mathematical analysis]]
[[Category:Differential calculus]]
[[Category:Functions and mappings]]
[[Category:Linear operators in calculus]]

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