导数与微分

在微分学中，函数的导数和微分关系密切，但意义却截然不同，用来表示与可微函数有关的两个重要数学对象。

什么是导数？

函数的导数测量函数值随输入变化而变化的速率。在多变量函数中，函数值的变化取决于自变量值变化的方向。因此，在这种情况下，选择一个特定的方向，并在该特定方向上区分功能。这个导数叫做方向导数。偏导数是一类特殊的方向导数。

Derivative of a vector-valued function f can be defined as the limit wherever it exists finitely. As mentioned before, this gives us the rate of increase of the function f along the direction of the vector u. In the case of a single-valued function, this reduces to the well-known definition of the derivative,  

For example, is everywhere differentiable, and the derivative is equal to the limit, , which is equal to . The derivatives of functi*** such as   exist everywhere. They are respectively equal to the functi*** .                                                                                

This is known as the first derivative. Usually the first derivative of function f is denoted by f ⁽¹⁾. Now using this notation, it is possible to define higher order derivatives. is the second order directional derivative, and denoting the n^th derivative by f ⁽ⁿ⁾ for each n, ,  defines the n^th derivative. 

什么是微分？

Differential of a function represents the change in the function with respect to changes in the independent variable or variables. In the usual notation, for a given function f of a single variable x, the total differential of order 1 df is given by, . This means that for an infinitesimal change in x(i.e. dx), there will be a  f ⁽¹⁾(x)dx change in f.

使用限制可以得到如下定义。假设∆x是x在任意点x的变化，∆f是函数f的相应变化。可以证明∆f=f（1）（x）∆x+ϵ，其中ϵ是误差。现在，极限∆x→0∆f/∆x=f（1）（x）（使用前面所述的导数定义），因此∆x→0ϵ/∆x=0。因此，可以得出结论，∆x→0ϵ=0。现在，用∆x→0∆f表示df，∆x→0∆x表示dx，严格地得到了微分的定义。

For example, the differential of the function is .

In the case of functi*** of two or more variables, the total differential of a function is defined as the sum of differentials in the directi*** of each of the independent variables. Mathematically, it can be stated as .