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The symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [1] [2]: 6 Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
The Gaussian function is the archetypal example of a bell shaped function. A bell-shaped function or simply 'bell curve' is a mathematical function having a characteristic "bell"-shaped curve.
Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. For unit variance, the n-th derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory.
The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions , which are smooth and certainly ...
More generally, its n th derivative is () = (), where is the n th (probabilist) Hermite polynomial. [ 24 ] The probability that a normally distributed variable X {\textstyle X} with known μ {\textstyle \mu } and σ 2 {\textstyle \sigma ^{2}} is in a particular set, can be calculated by using the fact that the fraction Z = ( X − μ ...
A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point [1] [2] and exactly one inflection point. Properties
The word derivative sounds fancy and perhaps a little intimidating. But the key thing to know about derivatives is that they are a financial contract whose value is derived from the value of ...
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.