# 隐函数

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 例子

### 反函数

x = f(y)

x表示y。这个解是

$y = f^{-1}(x).$

$R(x,y) = x-f(y) = 0. \,$

1. 对数函数 ln(x) 给出方程xey = 0或等价的x = ey的解 y = ln(x) . 这里 f(y) = ey 并且 f−1(x) = ln(x).
2. The product log is an implicit function giving the solution for y of the equation xy ey = 0.

### 代数函数

$a_n(x)y^n+a_{n-1}(x)y^{n-1}+\cdots+a_0(x)=0 \,$

where the coefficients ai(x) are polynomial functions of x. Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the unit circle equation:

$x^2+y^2-1=0. \,$

Solving for y gives an explicit solution:

$y=\pm\sqrt{1-x^2}. \,$

But even without specifying this explicit solution, it is possible to refer to the implicit solution of the unit circle equation.

While explicit solutions can be found for equations that are quadratic, cubic, and quartic in y, the same is not in general true for quintic and higher degree equations, such as

$y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0. \,$

Nevertheless, one can still refer to the implicit solution y = g(x) involving the multi-valued implicit function g.

## 隱函數的导数

• 隐函数左右两边对$x$求导（但要注意把$y$看作$x$的函数）；
• 利用一阶微分形式不变的性质分别对$x$$y$求导，再通过移项求得$\frac {dy}{dx}$的值；
• 把n元隐函数看作(n+1)元函数，通过多元函数偏导数的商求得n元隐函数的导数。举个例子，若欲求$z=f(x,y)$的导数$\frac {dy}{dx}$，那么可以将原隐函数通过移项化为$f(x,y,z)=0$的形式，然后通过$\frac {dy}{dx} = -\frac{F'_x}{F'_y}$（式中$F'_y$$F'_x$分别表示$y$$x$$z$的偏导数）来求解。

### 示例

• 針對$y^n$

$\frac{d}{dx}y^n = n \cdot y^{n-1}\frac{dy}{dx}$

• 針對$x^m y^n$

$\frac{d}{dx}x^m y^n = n \cdot x^m y^{n-1}\frac{dy}{dx} + m \cdot x^{m-1} y^n$

• $\ 12x^7-7x^4 y^3+6xy^5-14y^6+25=10$對x的導數。

${\color{Blue}12x^7}{\color{Red}-7x^4 y^3}{\color{Green}+6xy^5}{\color{Brown}-14y^6}+25=10$

1.兩邊皆取其相應的導數，得出

${\color{Blue}12\cdot7x^6}{\color{Red}-7\left(3x^4 y^2\frac{dy}{dx} + 4x^3 y^3 \right)}{\color{Green}+6\left(5xy^4\frac{dy}{dx} + y^5\right)}{\color{Brown}-14\cdot 6y^5\frac{dy}{dx}}+0=0$

2.移項處理。

${\color{Blue}84x^6}{\color{Red}- 28x^3 y^3}{\color{Green}+ 6y^5}={\color{Red}21x^4 y^2\frac{dy}{dx}}{\color{Green}- 30xy^4\frac{dy}{dx}}{\color{Brown}+84y^5\frac{dy}{dx}}$

3.抽出導數因子。

${\color{Blue}84x^6}{\color{Red}- 28x^3 y^3}{\color{Green}+ 6y^5}=\left({\color{Red}21x^4 y^2}{\color{Green}- 30xy^4}{\color{Brown}+84y^5} \right)\left( \frac{dy}{dx} \right)$

4.移項處理。

$\frac{dy}{dx} = \frac{{\color{Blue}84x^6}{\color{Red}- 28x^3 y^3}{\color{Green}+ 6y^5}}{{\color{Red}21x^4 y^2}{\color{Green}- 30xy^4}{\color{Brown}+84y^5}}$

5.完成。得出其導數為$\frac{84x^6 - 28x^3 y^3 + 6y^5}{21x^4 y^2 - 30xy^4 + 84y^5}$

6.選擇性步驟：因式分解處理。

$\frac{dy}{dx} = \frac{2\left(42x^6 - 14x^3 y^3 + 3y^5 \right)}{3y^2\left(7x^4 - 10xy^2 + 28y^3\right)}$