Generalizations of the derivative

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 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 f : R → R {displaystyle f:mathbb {R} o 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 L ( z ) = f ′ ( x ) z − f ′ ( x ) x + f ( x ) {displaystyle L(z)=f'(x)z-f'(x)x+f(x)} is tangent to the original function at the point ( x , f ( x ) ) , {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 L ( z ) = f ′ ( x ) z {displaystyle L(z)=f'(x)z} , L(z) becomes an actual linear operator on R considered as a vector space over itself. This motivates the following generalization to functions mapping Rm to Rn: ƒ is differentiable at x if there exists a linear operator A(x) (depending on x) such that Although this definition is perhaps not as explicit as the above, if such an operator exists, then it is unique, and in the one-dimensional case coincides with the original definition. (In this case the derivative is represented by a 1-by-1 matrix consisting of the sole entry f'(x).) Note that, in general, we concern ourselves mostly with functions being differentiable in some open neighbourhood of x {displaystyle x} rather than at individual points, as not doing so tends to lead to many pathological counterexamples. An n by m matrix, of the linear operator A(x) is known as Jacobian matrix Jx(ƒ) of the mapping ƒ at point x. Each entry of this matrix represents a partial derivative, specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobianmatrix of the composition g°f is a product of corresponding Jacobian matrices: Jx(g°f) =Jƒ(x)(g)Jx(ƒ). This is a higher-dimensional statement of the chain rule. For real valued functions from Rn to R (scalar fields), the total derivative can be interpreted as a vector field called the gradient. An intuitive interpretation of the gradient is that it points 'up': in other words, it points in the direction of fastest increase of the function. It can be used to calculate directional derivatives of scalar functions or normal directions.