微分与链式法则

微分

函数$f(x,y,z)$，有

$$\text{d}f=f_x\text{d}x+f_y\text{d}y+f_z\text{d}z$$$$\text{d}f=\frac{\partial f}{\partial x}\text{d}x+\frac{\partial f}{\partial y}\text{d}y+\frac{\partial f}{\partial z}\text{d}z$$

链式法则

当函数$f(x,y,z)$转换成看做是关于$t$的函数$f(t)$时

$x=x(t)$
$y=y(t)$
$z=z(t)$

有链式法则：

$$\frac{\text{d}f}{\text{d}t}=f_x\frac{\text{d}x}{\text{d}t}+f_y\frac{\text{d}y}{\text{d}t}+f_z\frac{\text{d}z}{\text{d}t}$$

注：

1. 在一元的课程中的链式法则，就是函数$f(x)$转换成关于$t$的函数$f(x(t))$
2. 一元导数是全导数，多元是偏导数

一切皆为链式法则

一元加法法则

$f=(u+v)$；$u=u(t)$；$v=v(t)$

$$\begin{array}{rl}\frac{\text{d}(u+v)}{\text{d}t}& =f_u\frac{\text{d}u}{\text{d}t}+f_v\frac{\text{d}v}{\text{d}t}\\ & =1\cdot u^{\prime }+1\cdot v^{\prime }\\ & =u^{\prime }+v^{\prime }\end{array}$$

一元乘法法则

$f=uv$；$u=u(t)$；$v=v(t)$

$$\begin{array}{rl}\frac{\text{d}(uv)}{\text{d}t}& =f_u\frac{\text{d}u}{\text{d}t}+f_v\frac{\text{d}v}{\text{d}t}\\ & =v\frac{\text{d}u}{\text{d}t}+u\frac{\text{d}v}{\text{d}t}\\ & =u^{\prime }v+u{v}^{\prime }\end{array}$$

一元除法法则

$f=\frac{u}{v}$；$u=u(t)$；$v=v(t)$

$$\begin{array}{rl}\frac{\text{d}(\frac{u}{v})}{\text{d}t}& =f_u\frac{\text{d}u}{\text{d}t}+f_v\frac{\text{d}v}{\text{d}t}\\ & =\frac{1}{v}\frac{\text{d}u}{\text{d}t}+\frac{-u}{v^2}\frac{\text{d}v}{\text{d}t}\\ & =\frac{u^{\prime }}{v}-\frac{u{v}^{\prime }}{v^2}\\ & =\frac{u^{\prime }v-u{v}^{\prime }}{v^2}\end{array}$$