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However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of f may be written as f ( − 1 ) ( x ) {\displaystyle f^{(-1)}(x)} for the first integral (this is easily confused with the inverse function f − 1 ( x ) {\displaystyle f ...
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 = = =.
For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +. The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts.
Higher-order methods for approximating the derivative, as well as methods for higher derivatives, exist. ... For example, [5] the first derivative can be calculated ...
Informally, this motivates Leibniz's notation for higher-order derivatives () =. When the independent variable x itself is permitted to depend on other variables, then the expression becomes more complicated, as it must include also higher order differentials in x itself.
The most common differential operator is the action of taking the derivative. Common notations for taking the first derivative with respect to a variable x include: , , , and . When taking higher, nth order derivatives, the operator may be written:
For example, the second order partial derivatives of a scalar function of n variables can be organized into an n by n matrix, the Hessian matrix. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a ...