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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 ...
, a branch of mathematics, the third derivative is the rate at which the
second derivative 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, ...
, or the rate of change of the rate of change, is changing. The third 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 ...
y = f(x) can be denoted by :\frac,\quad f(x),\quad\text\frac
(x) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, m ...
Other notations can be used, but the above are the most common.


Mathematical definitions

Let f(x) = x^4. Then f'(x) = 4x^3 and f''(x) = 12x^2. Therefore, the third derivative of ''f'' is, in this case, : f(x) = 24x or, using
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 ...
, : \frac ^4= 24x. Now for a more general definition. Let ''f'' be any function of ''x'' such that ''f'' ′′ is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
. Then the third derivative of ''f'' is given by : \frac
(x) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, m ...
= \frac ''(x) The third derivative is the rate at which the
second derivative 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, ...
(''f''′′(''x'')) is changing.


Applications in geometry

In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
, the
torsion of a curve In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a ...
— a fundamental property of curves in three dimensions — is computed using third derivatives of coordinate functions (or the position vector) describing the curve.


Applications in physics

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, particularly
kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
, jerk is defined as the third derivative of the position function of an object. It is, essentially, the rate at which
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
changes. In mathematical terms: : \mathbf(t)=\frac where j(''t'') is the jerk function with respect to time, and r(''t'') is the position function of the object with respect to time.


Economic examples

When campaigning for a second term in office, U.S. President
Richard Nixon Richard Milhous Nixon (January 9, 1913April 22, 1994) was the 37th president of the United States, serving from 1969 to 1974. A member of the Republican Party, he previously served as a representative and senator from California and was ...
announced that the rate of increase of inflation was decreasing, which has been noted as "the first time a sitting president used the third derivative to advance his case for reelection." Since
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reducti ...
is itself a derivative—the rate at which the purchasing power of money decreases—then the rate of increase of inflation is the derivative of inflation, opposite in sign to the second time derivative of the purchasing power of money. Stating that a function is
decreasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
is equivalent to stating that its derivative is negative, so Nixon's statement is that the second derivative of inflation is negative, and so the third derivative of purchasing power is positive. Nixon's statement allowed for the rate of inflation to increase, however, so his statement was not as indicative of stable prices as it sounds. Intergovernmental Panel on Climate Change published Summary for Policymakers where they state that growth of emissions have slowed. As emissions are the derivative of cumulative in the atmosphere, rate of change of their change is a third derivative.https://report.ipcc.ch/ar6wg3/pdf/IPCC_AR6_WGIII_HeadlineStatements.pdf


See also

* Aberrancy (geometry) *
Derivative (mathematics) 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 ...
*
Second derivative 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, ...


References

{{reflist Differential calculus