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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 = = =.
If f is a function, then its derivative evaluated at x is written ′ (). It first appeared in print in 1749. [3] Higher derivatives are indicated using additional prime marks, as in ″ for the second derivative and ‴ for the third derivative. The use of repeated prime marks eventually becomes unwieldy.
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
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 ...
A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h, h represents a small change in x, and it can be either positive or negative. The slope of this line is (+) ().
In this case one needs to consider the Fréchet derivative or Gateaux derivative. Example The five terms in the following expression correspond in the obvious way to the five partitions of the set { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} , and in each case the order of the derivative of f {\displaystyle f} is the number of parts in the partition:
Derivative Accuracy −5 −4 −3 −2 −1 0 1 2 3 4 5 1 2 −1/2: 0: 1/2: 4 1/12: −2/3: 0: 2/3: −1/12: 6 −1/60: 3/20: −3/4: 0: 3/4: −3/20: 1/60: 8 1/280 ...
These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. [1] The directional derivative provides a systematic way of finding these derivatives. [2]