📚 Implicit Differentiation: Core Techniques | 隐函数微分法核心技巧
Implicit differentiation is a powerful technique used when a curve is given by an equation that mixes both x and y, rather than by an explicit formula y=f(x). Instead of solving for y first, we differentiate every term with respect to x, treating y as a function of x, and then solve for dydx.
隐函数微分法是处理曲线方程中 x 与 y 混合出现、而不是写成显式 y=f(x) 时的一种强大工具。我们不必先解出 y，而是将每一项对 x 求导，把 y 看作 x 的函数，再解出 dydx。
1. What is an Implicit Function? | 什么是隐函数？
An implicit function is a relation between x and y written in the form F(x,y)=0, such as x2+y2=25, xy=sin x, or x3+y3=6xy.
隐函数是形如 F(x,y)=0 的 x 与 y 之间的关系式，例如 x2+y2=25、xy=sin x 或 x3+y3=6xy。
In many cases, solving for y explicitly is difficult or impossible, yet dydx can still be found using implicit differentiation.
在许多情况下，显式解出 y 很困难甚至不可能，但我们仍然可以利用隐函数微分法求出 dydx。
2. Why Differentiate Implicitly? | 为什么需要隐函数微分法？
Some curves, such as circles or ellipses, can be solved for y, but the result often involves square roots and requires choosing a branch. Differentiating the implicit equation directly is often faster and gives a single expression for the slope.
有些曲线如圆或椭圆可以通过解方程得到 y，但结果常含平方根且必须选择分支。直接对隐式方程求导通常更快，并且能得到斜率的统一表达式。
Other curves, such as xy = yx or x2+xy+y2=4, are extremely difficult or impossible to write as y=f(x) using elementary functions. Implicit differentiation becomes an essential strategy.
另一些曲线，如 xy = yx 或 x2+xy+y2=4，用初等函数写成 y=f(x) 极其困难甚至不可能。此时隐函数微分法便成为必要策略。
3. Core Technique: Differentiate Every Term with Respect to x | 核心技巧：对每一项关于 x 求导
The central idea is simple: differentiate both sides of the equation term by term with respect to x, and whenever you differentiate a term containing y, apply the chain rule.
核心思想很简单：对方程两边逐项对 x 求导，只要遇到含 y 的项，就使用链式法则。
Since y is treated as a function of x, the derivative of y itself is written as dydx. For example, ddx(y3)=3y2dydx.
因为把 y 看成 x 的函数，所以 y 本身的导数写作 dydx。例如，ddx(y3)=3y2dydx。
After differentiation, collect all terms containing dydx on one side and solve for dydx.
求导后，把所有含 dydx 的项移到一边，再解出 dydx。
4. Handling Powers and Composites of y | 处理 y 的幂和复合函数
For a power yn, the rule is:
对于幂函数 yn，规则是：
ddx(yn)=n yn-1dydx
For a composite function such as sin(y), ln(y), or ey, multiply the outer derivative by dydx:
对于复合函数如 sin(y)、ln(y) 或 ey，在求外层导数后要乘上 dydx：
- ddx[sin(y)]=cos(y)dydx
- ddx[ln(y)]=1ydydx
- ddx[ey]=eydydx
This is the most important habit to develop: never forget the extra factor dydx when differentiating an expression that contains y.
这是最需要养成的习惯：对含 y 的表达式求导时，切勿忘记额外因子 dydx。
5. Product Rule and Quotient Rule | 乘积法则与商法则
If a term is a product or quotient of functions of x and y, use the product or quotient rule while remembering the chain rule for the y-parts.
如果某项是 x 与 y 的函数相乘或相除，在使用乘积法则或商法则的同时，不要忘记对含 y 的部分使用链式法则。
For example, differentiate x2 y:
例如，对 x2 y 求导：
2xy + x2dydx
For xy, use the quotient rule:
对于 xy，使用商法则：
1· y - xdydx⁄y2
When the equation contains many terms, differentiate each separately and then rearrange.
当方程包含许多项时，先逐项求导，再整理合并。
6. Solving for dydx | 求解 dydx
After differentiating, the resulting equation will contain ordinary terms and terms multiplied by dydx. Factor out dydx and isolate it.
求导后，得到的方程会包含普通项和乘以 dydx 的项。提取公因子 dydx 并分离它。
Example: for x2+y2=25, differentiating gives 2x+2ydydx=0. Therefore:
例：对 x2+y2=25 求导得 2x+2ydydx=0。因此：
dydx=-xy
This formula gives the slope of the tangent at any point (x,y) on the circle (provided y ≠ 0).
该公式给出圆上任意点 (x,y) 处切线的斜率（前提是 y ≠ 0）。
7. Worked Examples | 典型例题
Example 1: Differentiate x2+xy+y2=4 implicitly.
例 1：对 x2+xy+y2=4 隐式求导。
Using the product rule on xy:
对 xy 使用乘积法则：
2x + y + xdydx + 2ydydx = 0
Collecting terms:
合并同类项：
(x+2y)dydx = -2x - y ⇒ dydx = -2x-yx+2y
Example 2: Find the slope of the tangent to x2+y2=25 at (3,4).
例 2：求圆 x2+y2=25 在点 (3,4) 处切线的斜率。
Using dydx=-x/y, at (3,4) we get:
利用 dydx=-x/y，在 (3,4) 处得到：
dydx=-34
8. Second Derivatives | 二阶导数
To find d2y⁄dx2, differentiate dydx implicitly again, treating it as a function of x and y, and then substitute known expressions.
求 d2y⁄dx2 时，再次对 dydx 作隐式求导，把它看作 x 和 y 的函数，然后代入已知表达式。
For x2+y2=25, we have dydx=-x/y. Differentiating again:
对于 x2+y2=25，已知 dydx=-x/y。再次求导：
d2y⁄dx2 = -y - xdydx⁄y2
Substitute dydx=-x/y:
代入 dydx=-x/y：
d2y⁄dx2 = -y + x2/y⁄y2 = -x2+y2⁄y3 = -25⁄y3
This method works even when the original equation cannot be solved explicitly for y.
即使原方程无法显式解出 y，这种方法依然有效。
9. Implicit vs Explicit Differentiation | 隐式微分与显式微分的对比
| Explicit | Implicit |
| y=f(x) | F(x,y)=0 |
| Differentiate y normally | Differentiate all terms; apply chain rule to y |
| Result is directly a function of x | Result often contains both x and y |
The explicit approach may be impossible for many curves, while implicit differentiation works directly on the equation.
显式方法在许多曲线上可能无法实现，而隐式微分法可以直接作用于方程本身。
10. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Mistake 1: Forgetting to multiply by dydx when differentiating yn or another function of y.
错误 1：对 yn 或其他含 y 的函数求导时忘记乘以 dydx。
Mistake 2: Treating y as a constant when differentiating terms like xy. Remember to use the product rule.
错误 2：求 xy 这类项时将 y 当成常数。记住要使用乘积法则。
Mistake 3: Forgetting to substitute given coordinates after solving for dydx. The formula is valid on the curve; plug in both x and y values.
错误 3：解出 dydx 后忘记代入给定坐标。该公式仅在曲线上成立，要同时代入 x 和 y 的值。
Mistake 4: Making algebraic errors when collecting terms with dydx. Carefully factor and isolate.
错误 4：在合并含 dydx 的项时出现代数错误。细心提取公因子并分离。
11. Practice Tips | 练习建议
Start with simple equations such as x2+y2=1 and y2=x3, then move to products and quotients like xy+sin y=1 or x3+y3=6xy.
从简单的方程如 x2+y2=1 和 y2=x3 开始，再过渡到乘积和商的形式，如 xy+sin y=1 或 x3+y3=6xy。
Always write down the chain rule step for each y-term. This reduces careless mistakes and makes your reasoning clear to examiners.
对每个含 y 的项都明确写出链式法则步骤。这能减少粗心错误，也让阅卷者看清你的思路。
Finally, check your answer by testing a point on the curve, or by solving for y explicitly when possible and differentiating normally.
最后，通过曲线上的一个点检验答案，或者在可能时先显式解出 y、用常规方法求导进行对照。
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