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Since the exponential function equals its derivative, this implies that the exponential function is monotonically increasing. Extension of exponentiation to positive real bases: Let b be a positive real number. The exponential function and the natural logarithm being the inverse each of the other, one has = ().
The exponential function is the unique function f ... = f(x) follows from the condition that f′(0) = 1 and the definition of the derivative as follows: ...
The exponential function () = is the unique differentiable function of a complex variable for which the derivative equals the function = and () = Power series definition [ edit ]
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.
For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series. Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in ...
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.
Toggle Derivatives of exponential and logarithmic functions subsection. 3.1 Logarithmic derivatives. ... The derivative of the function () = (() ...
The inverse function theorem together with the derivative of the exponential map provides information about the local behavior of exp. Any C k, 0 ≤ k ≤ ∞, ω map f between vector spaces (here first considering matrix Lie groups) has a C k inverse such that f is a C k bijection in an open set around a point x in the domain provided df x is ...