微分與微分
在微分學中,導數和微分關係密切,但又有很大的不同,用來表示與函數有關的兩個重要數學概念。
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.
什麼是差異化?
微分是求可微函數導數的過程。用D表示的D-算子在某些情況下表示微分。如果x是自變量,那麼D≡D/dx。D-算子是一個線性算子,即對於任意兩個可微函數f和g以及常數c,以下性質成立。
一、 D(f+g)=D(f)+D(g)
二。D(cf)=cD(f)
使用D-算子,與微分相關的其他規則可以表示如下。D(f g)=D(f)g+f D(g),D(f/g)=[D(f)g–f D(g)]/g2和D(f o g)=(D(f)o g)D(g)。
例如,當使用給定的規則將F(x)=x2sin x與x相異時,答案將是2xsin x+x2cosx。
微分和導數有什麼區別?•導數是指函數的變化率•微分是求函數導數的過程。 |