📚 AP Calculus: Derivatives | AP 微积分：导数
The derivative is the heartbeat of AP Calculus, capturing the instantaneous rate of change of a function. It reveals how a quantity varies at a single moment, providing the slope of a tangent line and driving applications from motion to optimisation. Mastery of differentiation rules, implicit techniques, and the analysis of function behaviour is essential for the AP exam.
导数是 AP 微积分的心脏，捕捉函数在某一时刻的瞬时变化率。它揭示了一个量在单个瞬间的变化，给出切线的斜率，并推动从运动到优化等一系列应用。掌握求导法则、隐函数技巧以及函数行为的分析，对 AP 考试至关重要。
1. Definition of the Derivative | 导数的定义
The derivative of a function f at a point x, denoted f ‘(x), is the limit of the difference quotient:
f ‘(x) = limh→0 (f(x+h) – f(x)) / h
函数 f 在点 x 处的导数记为 f ‘(x)，定义为差商的极限：
f ‘(x) = limh→0 (f(x+h) – f(x)) / h
Equivalently, the derivative at x = a can be expressed as f ‘(a) = limx→a (f(x) – f(a)) / (x – a). This limit, if it exists, represents the slope of the tangent line to the graph of f at that point. A function is differentiable at a point if the derivative exists there; differentiability implies continuity, but continuity does not guarantee differentiability (e.g., at a corner, cusp, or vertical tangent).
等价地，在 x = a 处的导数可以表示为 f ‘(a) = limx→a (f(x) – f(a)) / (x – a)。如果该极限存在，它表示该点处函数图像切线的斜率。函数在一点可导意味着该点处导数存在；可导必定连续，但连续不一定可导（例如在尖角、尖点或垂直切线处）。
2. Basic Differentiation Rules | 基本求导法则
The following table lists the fundamental derivative formulas that must be memorised for the AP exam.
下表列出了 AP 考试必须熟记的基本导数公式。
| Function f(x) | Derivative f ‘(x) |
| Constant c | 0 |
| xⁿ (power rule) | n xⁿ⁻¹ |
| eˣ | eˣ |
| aˣ (a > 0) | aˣ ln a |
| ln x | 1 / x |
| logₐ x | 1 / (x ln a) |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| cot x | -csc² x |
| sec x | sec x tan x |
| csc x | -csc x cot x |
These rules handle constant multiples and sums: (c f)’ = c f ‘ and (f ± g)’ = f ‘ ± g ‘.
这些法则可以处理倍数与和差： (c f)’ = c f ‘ 且 (f ± g)’ = f ‘ ± g ‘.
3. Product and Quotient Rules | 乘积法则与商法则
When differentiating the product of two functions, use the product rule:
(f · g)’ = f ‘ · g + f · g ‘
对两个函数的乘积求导，使用乘积法则：
(f · g)’ = f ‘ · g + f · g ‘
For the quotient of two functions, apply the quotient rule:
(f / g)’ = (f ‘ · g – f · g ‘) / g²
对两个函数的商求导，应用商法则：
(f / g)’ = (f ‘ · g – f · g ‘) / g²
Remember to keep the order consistent: ‘low d-high minus high d-low over low squared.’ These rules are frequently tested on the AP exam, especially in combination with the chain rule.
记住保持顺序一致：“低乘高导 减 高乘低导 除以低平方”。这些法则在 AP 考试中经常与链式法则结合考查。
4. The Chain Rule | 链式法则
The chain rule differentiates composite functions y = f(g(x)). If y = f(u) and u = g(x), then
dy/dx = dy/du · du/dx
链式法则用于复合函数 y = f(g(x)) 求导。若 y = f(u) 且 u = g(x)，则
dy/dx = dy/du · du/dx
In Leibniz notation, differentiate the ‘outside’ function with respect to the inner function, then multiply by the derivative of the inner function. For example, d/dx [sin(x²)] = cos(x²) · 2x. The chain rule can be extended to multiple layers: if y = f(g(h(x))), then dy/dx = f ‘(g(h(x))) · g ‘(h(x)) · h ‘(x).
在莱布尼茨记法中，先对外层函数关于内层函数求导，再乘以内层函数的导数。例如，d/dx [sin(x²)] = cos(x²) · 2x。链式法则可以推广到多层复合：若 y = f(g(h(x)))，则 dy/dx = f ‘(g(h(x))) · g ‘(h(x)) · h ‘(x).
5. Implicit Differentiation | 隐函数求导
When a relationship between x and y is defined implicitly (e.g., x² + y² = 25), differentiate both sides with respect to x, treating y as a function of x. Each time a term involving y is differentiated, multiply by dy/dx. For instance, differentiating x² + y² = 25 gives 2x + 2y (dy/dx) = 0, so dy/dx = -x / y.
当 x 和 y 之间的关系以隐式给出（如 x² + y² = 25），对等式两边关于 x 求导，并将 y 视为 x 的函数。每当对含 y 的项求导时，需乘以 dy/dx。例如，对 x² + y² = 25 求导得到 2x + 2y (dy/dx) = 0，因此 dy/dx = -x / y。
Implicit differentiation is essential for finding derivatives of inverse functions and for related‑rates problems. It also allows us to differentiate equations that cannot be easily solved for y.
隐函数求导对于求反函数的导数以及相关变化率问题至关重要。它还能让我们对难以解出 y 的方程直接求导。
6. Higher-Order Derivatives | 高阶导数
The second derivative, denoted f ”(x) or d²y/dx², is the derivative of the first derivative. It measures the rate of change of the slope, providing information about concavity and acceleration. Higher derivatives follow: f ”'(x), f ⁽⁴⁾(x), and in general f ⁽ⁿ⁾(x). In mechanics, if s(t) is position, then s ‘(t) is velocity and s ”(t) is acceleration.
二阶导数记为 f ”(x) 或 d²y/dx²，是一阶导数的导数。它衡量斜率的变化率，提供关于凹凸性和加速度的信息。更高阶的导数依次为：f ”'(x), f ⁽⁴⁾(x) 以及一般形式 f ⁽ⁿ⁾(x)。在力学中，如果 s(t) 表示位移，则 s ‘(t) 是速度，s ”(t) 是加速度。
Be careful with notation: d²y/dx² is not the same as (dy/dx)². The AP exam may ask you to find a higher derivative from implicit or parametric equations.
注意符号：d²y/dx² 不同于 (dy/dx)²。AP 考试可能要求从隐函数或参数方程求出高阶导数。
7. Tangent and Normal Lines | 切线与法线
Given a function y = f(x), the line tangent to the curve at a point (a, f(a)) has slope m = f ‘(a) and equation y – f(a) = f ‘(a)(x – a). The normal line is perpendicular to the tangent, so its slope is -1 / f ‘(a) (provided f ‘(a) ≠ 0), and its equation is y – f(a) = (-1/f ‘(a))(x – a).
对于函数 y = f(x)，曲线在点 (a, f(a)) 处的切线斜率为 m = f ‘(a)，方程为 y – f(a) = f ‘(a)(x – a)。法线与切线垂直，因此其斜率为 -1 / f ‘(a)（假设 f ‘(a) ≠ 0），方程为 y – f(a) = (-1/f ‘(a))(x – a)。
If f ‘(a) = 0, the tangent is horizontal and the normal line is vertical (x = a). If the derivative is undefined (vertical tangent), the tangent is x = a and the normal is horizontal.
若 f ‘(a) = 0，则切线水平，法线垂直 (x = a)。若导数不存在（垂直切线），则切线为 x = a，法线水平。
8. Analyzing Function Behavior: Extrema and Inflection Points | 分析函数行为：极值与拐点
Critical points occur where f ‘(x) = 0 or f ‘(x) does not exist. To classify them, use the First Derivative Test: if f ‘ changes from positive to negative, the function has a local maximum; if from negative to positive, a local minimum. The Second Derivative Test states that if f ‘(c) = 0 and f ”(c) > 0, then f has a local minimum at c; if f ”(c) < 0, a local maximum.
临界点发生在 f ‘(x) = 0 或 f ‘(x) 不存在的点。要对其分类，可使用一阶导数测试：若 f ‘ 由正变负，则函数有局部最大值；若由负变正，则为局部最小值。二阶导数测试指出：若 f ‘(c) = 0 且 f ”(c) > 0，则 f 在 c 处为局部最小值；若 f ”(c) < 0，则为局部最大值。
Inflection points are where the concavity changes, i.e., f ”(x) changes sign. They occur at points where f ”(x) = 0 or f ”(x) is undefined, provided the sign of f ” actually switches. A sign‑chart analysis of f ‘ and f ” is a powerful tool for sketching graphs.
拐点是凹凸性发生变化之处，即 f ”(x) 变号的地方。它们出现在 f ”(x) = 0 或 f ”(x) 不存在的点，且前提是 f ” 确实变号。对 f ‘ 和 f ” 进行符号表分析是绘制图像的有力工具。
9. The Mean Value Theorem and Related Rates | 均值定理与相关变化率
The Mean Value Theorem (MVT) states that for a function f continuous on [a, b] and differentiable on (a, b), there exists at least one c in (a, b) such that f ‘(c) = (f(b) – f(a)) / (b – a). Geometrically, there is a point where the tangent is parallel to the secant line. MVT justifies key results like the Increasing/Decreasing Function Theorem and the Racetrack Principle.
均值定理 (MVT) 表明：若函数 f 在 [a, b] 上连续，在 (a, b) 上可导，则 (a, b) 内至少存在一点 c 使得 f ‘(c) = (f(b) – f(a)) / (b – a)。从几何上看，存在一点其切线平行于割线。MVT 为诸如增/减函数定理以及赛道原理等重要结论提供了依据。
Related rates problems involve finding the rate at which one quantity changes by relating it to other quantities whose rates are known. Steps: (1) identify given rates and unknown rate; (2) write an equation linking the variables; (3) differentiate with respect to time t using the chain rule; (4) substitute known values and solve for the desired rate. Common models include expanding circles, sliding ladders, and filling tanks.
相关变化率问题涉及通过将待求量与已知变化率的量关联起来，求出一个量的变化速率。步骤：(1) 识别已知速率和未知速率；(2) 写出联系各变量的方程；(3) 运用链式法则对时间 t 求导；(4) 代入已知数值并解出所求速率。常见模型包括膨胀的圆、滑动的梯子和灌水的容器。
10. Derivatives of Parametric and Polar Functions (BC) | 参数方程与极坐标的导数（BC 内容）
For a curve defined parametrically by x = f(t), y = g(t), the slope of the tangent is given by dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. The second derivative is d²y/dx² = [d(dy/dx)/dt] / (dx/dt). Notice that d²y/dx² ≠ (d²y/dt²) / (d²x/dt²).
对于由参数方程 x = f(t), y = g(t) 给出的曲线，其切线斜率由 dy/dx = (dy/dt) / (dx/dt) 给出，前提是 dx/dt ≠ 0。二阶导数为 d²y/dx² = [d(dy/dx)/dt] / (dx/dt)。注意 d²y/dx²
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