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This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let P and Q be differential operators (with coefficients that are differentiable sufficiently many times) and R = P ∘ Q . {\displaystyle R=P\circ Q.}
A coefficient is usually a constant quantity, but the differential coefficient of f is a constant function only if f is a linear function. When f is not linear, its differential coefficient is a function, call it f ′, derived by the differentiation of f, hence, the modern term, derivative. The older usage is now rarely seen.
Even if the gradient theorem (also called fundamental theorem of calculus for line integrals) has been proved for a differentiable (so looked as smooth) curve so far, the theorem is also proved for a piecewise-smooth curve since this curve is made by joining multiple differentiable curves so the proof for this curve is made by the proof per ...
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g, ′ = () ′ = (′ + ′ ), ...
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [ 1 ] If A is a differentiable map from the real numbers to n × n matrices, then
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 ).
This formula can be obtained by Taylor series expansion: (+) = + ′ ()! ″ ()! () +. The complex-step derivative formula is only valid for calculating first-order derivatives. A generalization of the above for calculating derivatives of any order employs multicomplex numbers , resulting in multicomplex derivatives.
The q-derivative of a function is defined by the formula () = () (). For x nonzero, if f is a differentiable function of x then in the limit as q → 1 we obtain the ordinary derivative, thus the q-derivative may be viewed as its q-deformation.