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Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
for the first derivative, for the second derivative, for the third derivative, and for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken.
The derivative of ′ is the second derivative, denoted as ″ , and the derivative of ″ is the third derivative, denoted as ‴ . By continuing this process, if it exists, the n {\displaystyle n} th derivative is the derivative of the ( n − 1 ) {\displaystyle (n-1)} th derivative or the derivative of order ...
Further time derivatives have also been named, as snap or jounce (fourth derivative), crackle (fifth derivative), and pop (sixth derivative). [12] [13] The seventh derivative is known as "Bang," as it is a logical continuation to the cycle. The eighth derivative has been referred to as "Boom," and the 9th is known as "Crash."
The rate of change of jerk, the fourth derivative of displacement is known as jounce. [11] The SI unit of jounce is m ⋅ s − 4 {\displaystyle \mathrm {m\cdot s^{-4}} } which can be pronounced as metres per quartic second .
The coefficients given in the table above correspond to the latter definition. The theory of Lagrange polynomials provides explicit formulas for the finite difference coefficients. [4] For the first six derivatives we have the following:
y = x 4 – x has a 2nd derivative of zero at point (0,0), but it is not an inflection point because the fourth derivative is the first higher order non-zero derivative (the third derivative is zero as well). Points of inflection can also be categorized according to whether f ' (x) is zero or nonzero.
The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of quartic , but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form