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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. For example, the second partial derivatives of a function f(x, y) are: [6]
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 = = =.
With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally: =: () ().. This formula can be used to derive a formula that computes the symbol of the composition of differential operators.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
In Leibniz's notation, this formula is written as: ... For the first and fourth quadrant ... Differentiable function – Mathematical function whose derivative exists;
where x is thought of as a function of a new variable u and the function y on the left is expressed in terms of x while on the right it is expressed in terms of u. If y = f(x) where f is a differentiable function that is invertible, the derivative of the inverse function, if it exists, can be given by, [21]
The complex-step derivative formula is only valid for calculating first-order derivatives. A generalization of the above for calculating derivatives of any order employs multicomplex numbers , resulting in multicomplex derivatives.