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The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators [1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives ...
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse operation. This mirrors the conventional way the related theorems are presented in modern basic ...
In the context of complex analysis, the derivative of at is defined to be [2] ′ = (). Superficially, this definition is formally analogous to that of the derivative ...
The derivative of the function given by () = + + is ′ = + () () + = + (). Here the second term was computed using the chain rule and the third term using the product rule. The known derivatives of the elementary functions , , (), (), and =, as well as the constant , were also used.
This concept of complex differentiability shares several properties with real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule. [4] A function is holomorphic on an open set if it is complex differentiable at every point of .
In complex analysis, the central objects of study are holomorphic functions, which are complex-valued functions on the complex numbers where the Fréchet derivative exists. In geometric calculus, the geometric derivative satisfies a weaker form of the Leibniz (product) rule. It specializes the Fréchet derivative to the objects of geometric ...