![4 fonctions du second degré](https://upload.wikimedia.org/wikipedia/commons/c/ce/4_fonctions_du_second_degr%C3%A9.svg)
In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, the second derivative, or the second order derivative, of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the object is changing with respect to time. In
Leibniz notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ...
:
:
where ''a'' is acceleration, ''v'' is velocity, ''t'' is time, ''x'' is position, and d is the instantaneous "delta" or change. The last expression
is the second derivative of position (x) with respect to time.
On the
graph of a function
In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset ...
, the second derivative corresponds to the
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
or
concavity
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, ...
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.
Second derivative power rule
The
power rule
In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated usin ...
for the first derivative, if applied twice, will produce the second derivative power rule as follows:
:
Notation
The second derivative of a function
is usually denoted
.
That is:
:
When using
Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ...
for derivatives, the second derivative of a dependent variable with respect to an independent variable is written
:
This notation is derived from the following formula:
:
Alternative notation
As the previous section notes, the standard Leibniz notation for the second derivative is
. However, this form is not algebraically manipulable. That is, although it is formed looking like a fraction of differentials, the fraction cannot be split apart into pieces, the terms cannot be cancelled, etc. However, this limitation can be remedied by using an alternative formula for the second derivative. This one is derived from applying the
quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
to the first derivative. Doing this yields the formula:
:
In this formula,
represents the differential operator applied to
, i.e.,
,
represents applying the differential operator twice, i.e.,
, and
refers to the square of the differential operator applied to
, i.e.,
.
When written this way (and taking into account the meaning of the notation given above), the terms of the second derivative can be freely manipulated as any other algebraic term. For instance, the inverse function formula for the second derivative can be deduced from algebraic manipulations of the above formula, as well as the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
for the second derivative. Whether making such a change to the notation is sufficiently helpful to be worth the trouble is still under debate.
Example
Given the function
:
the derivative of is the function
:
The second derivative of is the derivative of
, namely
:
Relation to the graph
Concavity
The second derivative of a function can be used to determine the concavity of the graph of .
A function whose second derivative is positive will be
concave up
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
(also referred to as convex), meaning that the
tangent line
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
will lie below the graph of the function. Similarly, a function whose second derivative is negative will be
concave down
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function f on an in ...
(also simply called concave), 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.
Second derivative test
The relation between the second derivative and the graph can be used to test whether a
stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" inc ...
for a function (i.e., a point where
) is a
local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
or a
local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
. 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
Limit or Limits may refer to:
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* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
for the second derivative:
:
The limit is called the
second symmetric derivative In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined asThomson, p. 1.
: \lim_ \frac.
The expression under the limit is sometimes called the symmetric difference quotient. A function is said ...
.
Note that the second symmetric derivative may exist even when the (usual) second derivative does not.
The expression on the right can be written as a
difference quotient
In single-variable calculus, the difference quotient is usually the name for the expression
: \frac
which when taken to the limit as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact ...
of difference quotients:
:
This limit can be viewed as a continuous version of the
second difference
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
for
sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
.
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
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
, 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 approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or a ...
s, the second derivative is related to the best
quadratic approximation
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the ...
for a function . This is the
quadratic function
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
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
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
for the function centered at .
Eigenvalues and eigenvectors of the second derivative
For many combinations of
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
explicit formulas for
eigenvalues and eigenvectors of the second derivative
Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and discrete cases. In the discrete case, the standard Central difference#Higher-order differences ...
can be obtained. For example, assuming