Third derivative

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Specialized

In calculus, a branch of mathematics, the third derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing, used to define aberrancy.^[1] The third derivative of a function y=f(x) can be denoted by

{\frac {d^{3}y}{dx^{3}}},\quad f'''(x),\quad {\text{or }}{\frac {d^{3}}{dx^{3}}}[f(x)].

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(x) is, in this case,

f'''(x)=24x

or, using Leibniz notation,

{\frac {d^{3}}{dx^{3}}}[x^{4}]=24x

Now for a more general definition. Let $f(x)$ be any function of x. Then the third derivative of $f(x)$ is given by the following:

{\frac {d^{3}}{dx^{3}}}[f(x)]={\frac {d}{dx}}[f''(x)]

The third derivative is the rate at which the second derivative (f''(x)) is changing.

Applications in geometry

In differential geometry, the torsion of a curve — a fundamental property of curves in three dimensions — is computed using third derivatives of coordinate functions (or the position vector) describing the curve.^[2]

Applications in physics

Main article: Jerk (physics)

In physics, particularly kinematics, jerk is defined as the third derivative of the position function of an object. It is basically the rate at which acceleration is changing. In mathematical terms:

{\mathbf {j}}(t)={\frac {d^{3}{\mathbf {r}}}{dt^{3}}}

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 example

U.S. President Richard Nixon, when campaigning for a second term in office 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."^[3] Since inflation 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, or the second derivative of the function of purchasing power of money with respect to time. Stating that a function is decreasing is equivalent to stating that its derivative is negative, so Nixon's statement is that the second derivative of inflation—or the third derivative of purchasing power—is negative.

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.

References

↑ Schot, Stephen (November 1978). "Aberrancy: Geometry of the Third Derivative". Mathematics Magazine. 5. 51: 259–275. doi:10.2307/2690245. JSTOR 2690245.
↑ do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7.
↑ Rossi, Hugo (October 1996). "Mathematics Is an Edifice, Not a Toolbox" (PDF). Notices of the American Mathematical Society. 43 (10): 1108. Retrieved 13 November 2012.

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