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If a∈I is a limit point of I∩ [a,∞) and I∩(–∞,a] and if f is left and right differentiable at a, then f is called semi-differentiable at a. If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a symmetric derivative, which equals the arithmetic mean ...
f ′ (x) A function f of x, differentiated once in Lagrange's notation. One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by Euler and just popularized by the former. In Lagrange's notation, a prime mark denotes a derivative.
v. t. e. In mathematics, the derivative is a fundamental tool that quantifies the sensitivity of 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.
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.
e. In calculus, the differential represents the principal part of the change in a function with respect to changes in the independent variable. The differential is defined by where is the derivative of f with respect to , and is an additional real variable (so that is a function of and ). The notation is such that the equation.
In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as: [1][2] {\displaystyle \lim _ {h\to 0} {\frac {f (x+h)-f (x-h)} {2h}}.} The expression under the limit is sometimes called the symmetric difference quotient. [3][4] A function is said to be symmetrically differentiable at a point x if ...
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
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