📚 A-Level Maths: Complete Guide to Differentiating Trigonometric Functions | A-Level 数学：三角函数求导法则全梳理
Differentiating trigonometric functions is a cornerstone of A-Level Mathematics. This skill is tested across pure mathematics, mechanics, and statistics, and it is essential for solving problems involving tangents, normals, stationary points, and rates of change. In this guide, we will systematically review every trigonometric derivative you need, explain the key rules for combining functions, and highlight common exam pitfalls to help you secure full marks.
三角函数求导是 A-Level 数学的基石。这一技能在纯数学、力学与统计中均有考查，也是解决切线、法线、驻点及变化率问题的基础。在本指南中，我们将系统梳理所有需要掌握的三角函数导数，讲解组合求导的关键法则，并指出常见易错点，帮你稳拿满分。
1. Radian Measure and Standard Derivatives | 弧度制与标准导数公式
All differentiation formulas for trigonometric functions are derived using radian measure. If angles are measured in degrees, an extra factor of π/180 appears in the derivative. In A-Level Maths, always assume radians unless a question explicitly states degrees.
所有三角函数求导公式均基于弧度制推导。若使用角度制，导数中会出现 π/180 的额外因子。在 A-Level 数学中，除非题目明确给出度数，否则均默认使用弧度。
The following table lists the standard derivatives you must know. The first three are essential for A-Level Maths; the last three are required for Further Maths.
下表列出必须掌握的标准导数公式。前三个是 A-Level 数学的核心要求，后三个适用于进阶数学。
| f(x) | f'(x) |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| sec x | sec x tan x |
| cosec x | -cosec x cot x |
| cot x | -cosec² x |
Memorise these formulas thoroughly. In particular, note the negative signs that appear in the derivatives of cos x, cosec x and cot x.
请熟记这些公式，特别注意 cos x、cosec x 和 cot x 的导数中的负号。
2. Derivatives of sin x and cos x from First Principles | 从定义推导 sin x 和 cos x 的导数
Using the limit definition of a derivative, f'(x) = limₕ→₀ [f(x+h) – f(x)]/h, we can derive the derivative of sin x. Start with the identity sin A – sin B = 2 cos((A+B)/2) sin((A-B)/2):
利用导数定义 f'(x) = limₕ→₀ [f(x+h) – f(x)]/h，我们可推导 sin x 的导数。使用恒等式 sin A – sin B = 2 cos((A+B)/2) sin((A-B)/2)：
d/dx (sin x) = limₕ→₀ [sin(x+h) – sin x] / h = cos x
The derivation relies on the fundamental limit limₕ→₀ (sin h)/h = 1. In a similar way, using the identity cos A – cos B = -2 sin((A+B)/2) sin((A-B)/2), we obtain:
该推导依赖关键极限 limₕ→₀ (sin h)/h = 1。类似地，利用恒等式 cos A – cos B = -2 sin((A+B)/2) sin((A-B)/2)，可得：
d/dx (cos x) = -sin x
These two special limits are worth remembering: limₓ→₀ sin x / x = 1 and limₓ→₀ (1 – cos x)/x = 0. They are frequently needed in first-principles questions.
这两个特殊极限值得牢记：limₓ→₀ sin x / x = 1 和 limₓ→₀ (1 – cos x)/x = 0。它们在定义求导题目中经常出现。
3. Derivatives of tan, sec, cosec and cot | tan、sec、cosec 和 cot 的导数
Since tan x = sin x / cos x, we can apply the quotient rule to derive its derivative:
因为 tan x = sin x / cos x，可用商法则推导其导数：
d/dx (tan x) = (cos² x + sin² x) / cos² x = 1 / cos² x = sec² x
For sec x, recall that sec x = 1/cos x. Using the chain rule or the quotient rule gives:
对于 sec x，回忆 sec x = 1/cos x。利用链式法则或商法则可得：
d/dx (sec x) = sec x tan x
Similarly, cosec x = 1/sin x and cot x = cos x/sin x produce:
同理，cosec x = 1/sin x 和 cot x = cos x/sin x 给出：
d/dx (cosec x) = -cosec x cot x, d/dx (cot x) = -cosec² x
These results are best memorised, as they appear frequently in integration and differential equation questions. Notice that the derivatives of co-functions (cos, cot, cosec) all carry a negative sign.
这些结果最好熟记，它们在积分和微分方程题中经常出现。注意所有余函数（cos、cot、cosec）的导数都带负号。
4. The Chain Rule with Trigonometric Functions | 链式法则与三角函数
The chain rule states d/dx f(g(x)) = f'(g(x)) · g'(x). For trigonometric functions, this leads to:
链式法则表明 d/dx f(g(x)) = f'(g(x)) · g'(x)。对于三角函数，它给出了：
d/dx sin(kx) = k cos(kx), d/dx cos(kx) = -k sin(kx), d/dx tan(kx) = k sec²(kx)
For example, differentiate y = sin(3x² + 1). Let u = 3x² + 1, then dy/dx = cos(u) · du/dx = 6x cos(3x² + 1).
例如，对 y = sin(3x² + 1) 求导。令 u = 3x² + 1，则 dy/dx = cos(u) · du/dx = 6x cos(3x² + 1)。
The chain rule also applies to powers of trig functions. For instance, y = sin³ x means y = (sin x)³. Using the chain rule: dy/dx = 3(sin x)² · cos x = 3 sin² x cos x.
链式法则也适用于三角函数的幂。例如 y = sin³ x 即 y = (sin x)³。利用链式法则：dy/dx = 3(sin x)² · cos x = 3 sin² x cos x。
Always multiply by the derivative of the inner function. A common mistake is forgetting this extra factor, especially when the argument is not just x.
务必乘以内部函数的导数。常见错误是忘记这个额外因子，尤其当变量不是 x 时。
5. The Product Rule with Trigonometric Functions | 积法则与三角函数
The product rule states d/dx (uv) = u’v + uv’. It is used when a trig function is multiplied by another function of x. Consider y = x² sin x:
积法则为 d/dx (uv) = u’v + uv’。当三角函数与另一个关于 x 的函数相乘时使用。以 y = x² sin x 为例：
dy/dx = 2x sin x + x² cos x
Here u = x², v = sin x, so u’ = 2x and v’ = cos x. The product rule gives the result directly.
这里 u = x²，v = sin x，于是 u’ = 2x，v’ = cos x。积法则直接给出结果。
Another typical example is y = eˣ cos x. Applying the rule: dy/dx = eˣ cos x – eˣ sin x = eˣ(cos x – sin x). When a product has three factors, apply the rule twice, or use the extended form.
另一个典型例子是 y = eˣ cos x。运用法则：dy/dx = eˣ cos x – eˣ sin x = eˣ(cos x – sin x)。若乘积含三个因子，可两次使用积法则，或使用扩展形式。
6. The Quotient Rule with Trigonometric Functions | 商法则与三角函数
The quotient rule states d/dx (u/v) = (u’v – uv’)/v². It is essential when a trig function is divided by another function, or vice versa. Consider y = (sin x)/x:
商法则为 d/dx (u/v) = (u’v – uv’)/v²。当三角函数除以另一个函数或反之时，该法则至关重要。以 y = (sin x)/x 为例：
dy/dx = (x cos x – sin x) / x²
For a question where the denominator is also a trig expression, such as y = (tan x)/(1 + cos x), let u = tan x and v = 1 + cos x. Then u’ = sec² x and v’ = -sin x. Applying the quotient rule:
当分母也是三角表达式时，例如 y = (tan x)/(1 + cos x)，令 u = tan x，v = 1 + cos x。则 u’ = sec² x，v’ = -sin x。应用商法则：
dy/dx = [sec² x (1 + cos x) – tan x(-sin x)] / (1 + cos x)²
Simplify carefully when possible. Remember that v² is always positive, but that does not mean the entire derivative is positive.
尽量仔细化简。注意 v² 总是正的，但这不代表整个导数为正。
7. Implicit Differentiation and Higher-Order Derivatives | 隐函数求导与高阶导数
When a trig function defines y implicitly, differentiate both sides with respect to x. For example, if sin y = x, then cos y · dy/dx = 1, so dy/dx = 1/cos y = sec y. This technique appears in both pure mathematics and mechanics problems.
当三角函数隐含定义 y 时，对两边关于 x 求导。例如若 sin y = x，则 cos y · dy/dx = 1，因此 dy/dx = 1/cos y = sec y。该技巧在纯数学和力学题中都会出现。
Higher-order derivatives of trig functions follow a repeating pattern. For y = sin x:
三角函数的高阶导数呈现循环规律。对于 y = sin x：
-
y’ = cos x
y’ = cos x
-
y” = -sin x
y” = -sin x
-
y”’ = -cos x
y”’ = -cos x
-
y”” = sin x
y”” = sin x
This cycle of four repeats indefinitely. It is especially useful when solving differential equations or classifying stationary points using the second derivative test.
这一四步循环无限重复。它在解微分方程或利用二阶导数判定驻点类型时非常有用。
8. Exam-Style Worked Examples | 考试典型例题
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