Differentiating Trigonometric Functions | 三角函数的微分

📚 Differentiating Trigonometric Functions | 三角函数的微分

Differentiating trigonometric functions is one of the most important topics in Edexcel A-Level Mathematics. It builds directly on the chain rule, product rule and quotient rule, and it is tested regularly in both pure mathematics and applied contexts such as rates of change and oscillatory motion.

三角函数的微分是 Edexcel A-Level 数学中最重要的主题之一。它直接建立在链式法则、乘积法则和商法则的基础上，并且在纯数学以及变化率和振动运动等应用情境中经常考查。

1. Standard Derivatives of sin x and cos x | 正弦与余弦的标准导数

The two fundamental derivatives are used so often that they must become automatic. They are given in the Edexcel formula booklet, but recalling them instantly will save time and reduce errors.

这两个基本导数使用非常频繁，必须达到自动化的程度。它们在 Edexcel 公式手册中有给出，但是能够立即回忆出来可以节省时间并减少错误。

d/dx (sin x) = cos x

d/dx (cos x) = -sin x

The negative sign in the derivative of cos x is the most common sign error in the whole topic. Since cos x is a decreasing function on (0, π), its gradient is negative there.

cos x 导数中的负号是整个主题中最常见的符号错误。因为 cos x 在区间 (0, π) 上是递减函数，所以其斜率在该区间内为负。

These results can also be interpreted graphically: the slope of the sine curve at any point equals the cosine value at that point, while the slope of the cosine curve is the negative sine value.

这些结果也可以从图形上解释：正弦曲线上任意一点的斜率等于该点处的余弦值，而余弦曲线上任意一点的斜率等于该点处正弦值的相反数。

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2. Derivative of tan x | 正切函数的导数

Since tan x = sin x / cos x, the derivative can be obtained from the quotient rule. The standard result is:

因为 tan x = sin x / cos x，其导数可以通过商法则得到。标准结果是：

d/dx (tan x) = sec² x

Here sec² x means (sec x)². Some students incorrectly write sec x², which is not the same thing.

这里的 sec² x 表示 (sec x)²。有些学生错误地写成 sec x²，那是不同的含义。

An equivalent form is 1 + tan² x because of the identity sec² x = 1 + tan² x. Both forms are acceptable in Edexcel markschemes.

由于恒等式 sec² x = 1 + tan² x，一个等价形式是 1 + tan² x。在 Edexcel 评分标准中两种形式都可以接受。

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3. Derivatives of sec x, cosec x and cot x | 正割、余割与余切函数的导数

Edexcel also tests the reciprocal trigonometric functions. The three standard results are:

Edexcel 也考查倒数三角函数。三个标准结果是：

d/dx (sec x) = sec x tan x

d/dx (cosec x) = -cosec x cot x

d/dx (cot x) = -cosec² x

Notice the pairing: sec x goes with tan x, and cosec x goes with cot x. The derivative of cot x is similar to that of tan x but negative and with cosec² x.

注意配对关系：sec x 与 tan x 配对，cosec x 与 cot x 配对。cot x 的导数与 tan x 的导数类似，但符号为负且使用 cosec² x。

These formulas can be derived from the reciprocal definitions using the quotient rule, but memorising them is expected for exam efficiency.

这些公式可以通过倒数定义使用商法则推导出来，但为了考试效率，熟记它们是必要的。

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4. Chain Rule with Trigonometric Functions | 三角函数的链式法则

When the angle is a function of x, such as ax + b, you must multiply by the derivative of the angle. The key results are:

当角度是 x 的函数时，例如 ax + b，你必须乘以角度的导数。关键结果是：

d/dx [sin(ax + b)] = a cos(ax + b)

d/dx [cos(ax + b)] = -a sin(ax + b)

d/dx [tan(ax + b)] = a sec²(ax + b)

Example: If y = sin(5x), then dy/dx = 5 cos(5x). The factor 5 comes from differentiating the inner function 5x.

例如：如果 y = sin(5x)，那么 dy/dx = 5 cos(5x)。系数 5 来自对内层函数 5x 求导。

For powers of trigonometric functions, such as sin³ x, write it as (sin x)³ and apply the chain rule twice: dy/dx = 3(sin x)² cos x.

对于三角函数的幂，例如 sin³ x，将其写作 (sin x)³ 并应用两次链式法则：dy/dx = 3(sin x)² cos x。

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5. Product Rule with Trigonometric Functions | 三角函数的乘积法则

When a trigonometric function is multiplied by another function, use the product rule. If y = u v, then dy/dx = u dv/dx + v du/dx.

当三角函数与另一个函数相乘时，使用乘积法则。如果 y = u v，那么 dy/dx = u dv/dx + v du/dx。

Example: Differentiate y = x² sin x. Let u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x, so dy/dx = 2x sin x + x² cos x.

例如：求 y = x² sin x 的导数。设 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x，因此 dy/dx = 2x sin x + x² cos x。

Another common type is y = eˣ cos x. Here dy/dx = eˣ cos x – eˣ sin x, which can be factorised as eˣ(cos x – sin x).

另一种常见类型是 y = eˣ cos x。此时 dy/dx = eˣ cos x – eˣ sin x，可以分解为 eˣ(cos x – sin x)。

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6. Quotient Rule with Trigonometric Functions | 三角函数的商法则

The quotient rule is required

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