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The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
In calculus, the product rule (or Leibniz rule [1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.For two functions, it may be stated in Lagrange's notation as () ′ = ′ + ′ or in Leibniz's notation as () = +.
This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable: [10] it is a more general one, since, as noted a by Henrici (1993, p. 294), the limit may exist for functions that are not even differentiable at =. [11] According to Fichera (1969, p. 28), the first to identify the areolar derivative ...
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
In the calculus of variations and classical mechanics, the Euler–Lagrange equations [1] are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or ...
The sign of the signed curvature is the same as the sign of the second derivative of f. If it is positive then the graph has an upward concavity, and, if it is negative the graph has a downward concavity. If it is zero, then one has an inflection point or an undulation point.
The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, ∇ u v {\displaystyle \nabla _{\mathbf {u} }{\mathbf {v} }} , which takes as its inputs: (1) a vector, u , defined at a point P , and (2) a vector field v defined in a ...