導數與微分
在微分學中，函數的導數和微分關係密切，但意義卻截然不同，用來表示與可微函數有關的兩個重要數學對象。
什麼是導數？
函數的導數測量函數值隨輸入變化而變化的速率。在多變量函數中，函數值的變化取決於自變量值變化的方向。因此，在這種情況下，選擇一個特定的方向，並在該特定方向上區分功能。這個導數叫做方向導數。偏導數是一類特殊的方向導數。
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 nth derivative by f (n) for each n, , defines the nth derivative.
什麼是微分？
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義。假設∆x是x在任意點x的變化，∆f是函數f的相應變化。可以證明∆f=f（1）（x）∆x+ϵ，其中ϵ是誤差。現在，極限∆x→0∆f/∆x=f（1）（x）（使用前面所述的導數定義），因此∆x→0ϵ/∆x=0。因此，可以得出結論，∆x→0ϵ=0。現在，用∆x→0∆f表示df，∆x→0∆x表示dx，嚴格地得到了微分的定義。
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 .
| 導數和微分的區別是什麼？•導數是指函數的變化率，而微分是指自變量發生變化時函數的實際變化。•導數由給出，微分由給出。 |
