In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as
The expression under the limit is sometimes called the symmetric difference quotient. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point.
If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f(x) = x, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient.
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.
Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
For the absolute value function , using the notation for the symmetric derivative, we have at that
Hence the symmetric derivative of the absolute value function exists at and is equal to zero, even though its ordinary derivative does not exist at that point (due to a "sharp" turn in the curve at ).
Note that in this example both the left and right derivatives at 0 exist, but they are unequal (one is −1, while the other is +1); their average is 0, as expected.
For the function , at we have
Again, for this function the symmetric derivative exists at , while its ordinary derivative does not exist at due to discontinuity in the curve there. Furthermore, neither the left nor the right derivative is finite at 0, i.e. this is an essential discontinuity.
The Dirichlet function, defined as
has a symmetric derivative at every , but is not symmetrically differentiable at any ; i.e. the symmetric derivative exists at rational numbers but not at irrational numbers.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In single-variable calculus, the difference quotient is usually the name for the expression which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x + h) - x = h in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h).
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative exists.
The goal of this course is to give an introduction to the theory of distributions and cover the fundamental results of Sobolev spaces including fractional spaces that appear in the interpolation theor
Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.
This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs.
Plant natural products (PNPs) and their derivatives are important but underexplored sources of pharmaceutical molecules. To access this untapped potential, the reconstitution of heterologous PNP biosynthesis pathways in engineered microbes provides a valua ...
We develop a novel 2D functional learning framework that employs a sparsity-promoting regularization based on second-order derivatives. Motivated by the nature of the regularizer, we restrict the search space to the span of piecewise-linear box splines shi ...
We study a variant of the interpolation problem where the continuously defined solution is regularized by minimizing the L p -norm of its second-order derivative. For this continuous-domain problem, we propose an exact discretization scheme that restricts ...