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The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
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 = = =.
Since the power-series coefficients of the exponential are well known, and higher-order derivatives of the monomial x n can be written down explicitly, this differential-operator representation gives rise to a concrete formula for the coefficients of H n that can be used to quickly compute these polynomials.
Their algorithm is applicable to higher-order derivatives. A method based on numerical inversion of a complex Laplace transform was developed by Abate and Dubner. [ 21 ] An algorithm that can be used without requiring knowledge about the method or the character of the function was developed by Fornberg.
In applied mathematics, the Ostrogradsky instability is a feature of some solutions of theories having equations of motion with more than two time derivatives (higher-derivative theories). It is suggested by a theorem of Mikhail Ostrogradsky in classical mechanics according to which a non-degenerate Lagrangian dependent on time derivatives ...
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
The order of the polynomial in the general smoothstep is 2n + 1. With n = 1, the slopes or first derivatives of the smoothstep are equal to zero at the left and right edge (x = 0 and x = 1), where the curve is appended to the constant or saturated levels.
Derivative = Antiderivative ... Higher order derivatives are given by () ... (equations 7.1.25–28). This allows one to choose the fastest approximation suitable ...