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velocity is the derivative (with respect to time) of an object's displacement (distance from the original position) acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position. For example, if an object's position on a line is given by
The derivative function becomes a map between the tangent bundles of and . This definition is used in differential geometry. [49] Differentiation can also be defined for maps between vector space, such as Banach space, in which those generalizations are the Gateaux derivative and the Fréchet derivative. [50]
The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point. Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain ...
The exterior derivative is a notion of differentiation of differential forms which generalizes the differential of a function (which is a differential 1-form). Pullback is, in particular, a geometric name for the chain rule for composing a map between manifolds with a differential form on the target manifold.
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as ...
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]
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if f {\displaystyle f} is a holomorphic function , real-valued on the real line, which can be evaluated at points in the complex plane near x {\displaystyle x} , then there are stable methods.