📚 IB Maths: Advanced Differentiation Techniques Masterclass | IB数学：微分进阶技巧全梳理
Differentiation is one of the most powerful tools in the IB Mathematics curriculum, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI) at Higher Level, and to a significant extent at Standard Level. While the basic rules — power, product, quotient, and chain — form the foundation, advanced differentiation techniques are what separate a good IB candidate from an excellent one. This guide consolidates every advanced differentiation technique you need, with clear explanations and IB-style examples.
微分是 IB 数学课程中最强大的工具之一，在分析与方法（AA）和应用与解释（AI）两个方向的高级水平（HL）中都是重点内容，在标准水平（SL）中也占据重要地位。基本规则——幂法则、乘积法则、商法则和链式法则——只是基础，真正让优秀 IB 考生脱颖而出的，是对进阶微分技巧的系统掌握。本文为你梳理所有必考的进阶微分技巧，并配以清晰的讲解和 IB 风格例题。
1. The Chain Rule in Depth | 链式法则的深入理解
The chain rule is arguably the most frequently tested differentiation rule in IB. It states that if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). In practice, you must learn to identify the ‘outer function’ and the ‘inner function’ quickly and reliably.
链式法则是 IB 考试中最常考的微分法则。它表明：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。在实际解题中，你必须学会快速而准确地识别“外层函数”和“内层函数”。
Consider y = sin(3x² + 4). The outer function is sin(u) and the inner function is u = 3x² + 4. Applying the chain rule:
考虑 y = sin(3x² + 4)。外层函数是 sin(u)，内层函数是 u = 3x² + 4。应用链式法则：
dy/dx = cos(3x² + 4) × (6x) = 6x·cos(3x² + 4)
A common IB examiner’s trap involves composite functions where the inner function itself requires the product or quotient rule. For example, y = ln(x²·eˣ). Here, the outer function is ln(u), but the inner function u = x²·eˣ must be differentiated using the product rule before the chain rule can be completed.
IB 考官常设置的陷阱是：复合函数的内层函数本身还需要乘积法则或商法则。例如，y = ln(x²·eˣ)。这里外层函数是 ln(u)，但内层函数 u = x²·eˣ 必须先使用乘积法则求导，才能完成链式法则。
2. Implicit Differentiation | 隐函数微分
When a relationship between x and y is given implicitly, such as x² + y² = 25, we cannot simply solve for y and differentiate. Instead, we differentiate both sides of the equation with respect to x, treating y as a function of x, and then solve for dy/dx.
当 x 和 y 之间的关系以隐式形式给出时，例如 x² + y² = 25，我们无法简单地解出 y 再求导。相反，我们对等式两边同时对 x 求导，将 y 视为 x 的函数，然后解出 dy/dx。
Differentiating x² + y² = 25 gives 2x + 2y·dy/dx = 0, hence dy/dx = -x/y. This technique is essential for finding gradients of curves that cannot be expressed as y = f(x), such as circles, ellipses, and more complex relations.
对 x² + y² = 25 两边求导得 2x + 2y·dy/dx = 0，因此 dy/dx = -x/y。这一技巧对于求无法表示为 y = f(x) 的曲线（如圆、椭圆及更复杂的关系）的斜率至关重要。
In IB exams, implicit differentiation often appears together with the equation of a tangent or normal. Remember that the normal gradient is the negative reciprocal of the tangent gradient, m_normal = -1/m_tangent.
在 IB 考试中，隐函数微分常与切线或法线的方程结合考查。切记：法线的斜率是切线斜率的负倒数，m_法线 = -1/m_切线。
3. Parametric Differentiation | 参数方程微分
When x and y are both given as functions of a parameter t, such as x = t² and y = t³, the derivative dy/dx is found by dividing dy/dt by dx/dt. The formula is:
当 x 和 y 都以参数 t 的函数形式给出时，如 x = t²，y = t³，导数 dy/dx 通过 dy/dt 除以 dx/dt 得到。公式为：
dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0
For x = t², y = t³, we have dy/dt = 3t² and dx/dt = 2t, so dy/dx = 3t/2. Note that the final expression is often given in terms of the parameter t, which is perfectly acceptable in IB unless otherwise stated.
对于 x = t²，y = t³，我们有 dy/dt = 3t²，dx/dt = 2t，所以 dy/dx = 3t/2。注意，最终表达式通常以参数 t 表示，除非题目另有要求，这在 IB 中是完全可接受的。
Higher-order derivatives can also be found using the parameter: d²y/dx² = d(dy/dx)/dt ÷ dx/dt. This is a subtle point that many students miss — you must divide by dx/dt again, not by dt/dx.
高阶导数也可以利用参数求出：d²y/dx² = d(dy/dx)/dt ÷ dx/dt。这是许多学生容易忽略的关键点——你必须再次除以 dx/dt，而不是除以 dt/dx。
4. Logarithmic Differentiation | 对数微分法
Logarithmic differentiation is a powerful technique for differentiating functions of the form y = [f(x)]^g(x), or products and quotients of many factors. The method involves taking the natural logarithm of both sides, simplifying using log laws, and then differentiating implicitly.
对数微分法是处理形如 y = [f(x)]^g(x) 的函数，或包含多个因子的乘积、商的高效技巧。方法是对等式两边取自然对数，利用对数运算法则化简，然后隐式求导。
Example: Differentiate y = xˣ. Taking ln of both sides gives ln y = x·ln x. Differentiating implicitly:
例：求 y = xˣ 的导数。两边取 ln 得 ln y = x·ln x。隐式求导：
(1/y)·dy/dx = ln x + 1, so dy/dx = xˣ(ln x + 1)
This technique is also extremely useful when differentiating expressions like y = x²·√(x+1)/(x-1)³, where direct application of product and quotient rules would be lengthy and error-prone.
当处理形如 y = x²·√(x+1)/(x-1)³ 的表达式时，直接应用乘积法则和商法则会非常冗长且易错，此时对数微分法就显示出极大的优势。
5. Derivatives of Inverse Trigonometric Functions | 反三角函数的导数
IB Analysis and Approaches HL requires knowledge of the derivatives of the six inverse trigonometric functions, though arcsin, arccos, and arctan are the most commonly tested.
IB 分析与方法 HL 要求掌握六个反三角函数的导数，其中 arcsin（反正弦）、arccos（反余弦）和 arctan（反正切）最为常考。
| Function | Derivative |
| y = arcsin(x) | dy/dx = 1/√(1-x²) |
| y = arccos(x) | dy/dx = -1/√(1-x²) |
| y = arctan(x) | dy/dx = 1/(1+x²) |
These derivatives are derived from the inverse function theorem. For example, for y = arctan(x), we know tan(y) = x. Differentiating both sides with respect to x gives sec²(y)·dy/dx = 1, hence dy/dx = 1/sec²(y) = 1/(1+tan²(y)) = 1/(1+x²).
这些导数由反函数定理推导得出。例如，对于 y = arctan(x)，我们有 tan(y) = x。对两边关于 x 求导得 sec²(y)·dy/dx = 1，所以 dy/dx = 1/sec²(y) = 1/(1+tan²(y)) = 1/(1+x²)。
When the argument is not simply x, remember to multiply by the derivative of the argument. For instance, d/dx[arcsin(2x)] = 2/√(1-4x²).
当自变量不是简单的 x 时，记得乘以自变量的导数。例如，d/dx[arcsin(2x)] = 2/√(1-4x²)。
6. Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The derivatives of eˣ and ln x are fundamental, but IB exams frequently test variations such as aˣ and logₐx. The general formulas are:
eˣ 和 ln x 的导数是基础，但 IB 考试经常考查 aˣ 和 logₐx 之类的变体。一般公式为：
d/dx(aˣ) = aˣ·ln a; d/dx(logₐx) = 1/(x·ln a)
For example, the derivative of 2ˣ is 2ˣ·ln 2, and the derivative of log₁₀(x) is 1/(x·ln 10). These formulas come from converting to base e: aˣ = e^(x·ln a) and logₐx = ln x / ln a.
例如，2ˣ 的导数是 2ˣ·ln 2，log₁₀(x) 的导数是 1/(x·ln 10)。这些公式源于换底公式：aˣ = e^(x·ln a)，logₐx = ln x / ln a。
A common application is differentiating y = e^(2x)·ln(3x). Using the product rule: dy/dx = 2e^(2x)·ln(3x) + e^(2x)·(1/x) = e^(2x)[2ln(3x) + 1/x].
一个常见应用是求 y = e^(2x)·ln(3x) 的导数。使用乘积法则：dy/dx = 2e^(2x)·ln(3x) + e^(2x)·(1/x) = e^(2x)[2ln(3x) + 1/x]。
7. Higher-Order Derivatives and Applications | 高阶导数及其应用
Higher-order derivatives, especially the second derivative, play a crucial role in curve sketching and optimization. The notation includes f”(x), d²y/dx², and y”. The second derivative tells us about concavity: if f”(x) > 0, the curve is concave up; if f”(x) < 0, it is concave down.
高阶导数，尤其是二阶导数，在曲线绘制和优化问题中起着关键作用。记号包括 f”(x)，d²y/dx² 和 y”。二阶导数告诉我们函数的凹凸性：若 f”(x) > 0，曲线向上凹；若 f”(x) < 0，曲线向下凹。
At a point where f'(x) = 0, the second derivative test determines the nature of the stationary point:
在 f'(
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