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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 symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied (the second partial derivatives are not continuous). The function f(x, y), as shown in equation , does not have symmetric second derivatives at its origin.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified ...
In linear algebra, the n-th symmetric power of a vector space V is the vector subspace of the symmetric algebra of V consisting of degree-n elements (here the product is a tensor product). In algebraic topology , the n -th symmetric power of a topological space X is the quotient space X n / S n {\displaystyle X^{n}/{\mathfrak {S}}_{n}} , as in ...
where the f k = f k (x 1, ... , x n) are functions of all the coordinates. A differential 1-form is integrated along an oriented curve as a line integral. The expressions dx i ∧ dx j, where i < j can be used as a basis at every point on the manifold for all 2-forms. This may be thought of as an infinitesimal oriented square parallel to the x ...
In calculus, the power rule is used to differentiate functions of the form () =, whenever is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule.
[1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups
In calculus, the reciprocal rule gives the derivative of the reciprocal of a function f in terms of the derivative of f.The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been established for positive exponents.