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In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and () The quotient rule states that the derivative of h(x) is
The total derivative / of p with respect to r, for example, gives the sign and magnitude of the reaction of the market price to the exogenous variable r. In the indicated system, there are a total of six possible total derivatives, also known in this context as comparative static derivatives : dp / dr , dp / dw , dp / dI , dq / dr , dq / dw ...
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.
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 classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
A graph edge represents a relationship between two entities, e.g. pairwise interactions or similarity based on comparisons of geometric neighborhoods (for example of pixels in images) or of another feature, with the edge weight encoding the strength of this relationship.
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 definitions are applied to graphs as follows. If a function (a -cochain) is defined at the nodes of a graph: ,,, … then its exterior derivative (or the differential) is the difference, i.e., the following function defined on the edges of the graph (-cochain):