Derivative (generalizations)

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.

In real, complex, and functional analysis, derivatives are generalized to functions of several real or complex variables and functions between topological vector spaces. An important case is the variational derivative in the calculus of variations. Repeated application of differentiation leads to derivatives of higher order and differential operators.

The derivative is often met for the first time as an operation on a single real function of a single real variable. One of the simplest settings for generalizations is to vector valued functions of several variables (most often the domain forms a vector space as well). This is the field of multivariable calculus.

In one-variable calculus, we say that a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is differentiable at a point x if the limit

exists. Its value is then the derivative ƒ'(x). A function is differentiable on an interval if it is differentiable at every point within the interval. Since the line ${\displaystyle L(z)=f'(x)z-f'(x)x+f(x)}$ is tangent to the original function at the point ${\displaystyle (x,f(x)),}$ the derivative can be seen as a way to find the best linear approximation of a function. If one ignores the constant term, setting ${\displaystyle L(z)=f'(x)z}$, L(z) becomes an actual linear operator on R considered as a vector space over itself.

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