微分与微分
在微分学中,导数和微分关系密切,但又有很大的不同,用来表示与函数有关的两个重要数学概念。
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 nth derivative by f (n) for each n, , defines the nth derivative.
什么是差异化?
微分是求可微函数导数的过程。用D表示的D-算子在某些情况下表示微分。如果x是自变量,那么D≡D/dx。D-算子是一个线性算子,即对于任意两个可微函数f和g以及常数c,以下性质成立。
一、 D(f+g)=D(f)+D(g)
二。D(cf)=cD(f)
使用D-算子,与微分相关的其他规则可以表示如下。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,当使用给定的规则将F(x)=x2sin x与x相异时,答案将是2xsin x+x2cosx。
微分和导数有什么区别?•导数是指函数的变化率•微分是求函数导数的过程。 |