📚 Example 6.4.1: Differentiating sin x and cos x | 例题 6.4.1：正弦与余弦函数的求导
In AQA A-Level Mathematics, trigonometric differentiation is a core skill that appears in both Pure Year 1 and Pure Year 2. This article works through Example 6.4.1, a typical question that asks you to differentiate a function involving sin x and cos x, evaluate the derivative at a given point, and find a tangent equation. We will explain each step in detail so that you can apply the same method to exam questions.
在 AQA A-Level 数学中，三角函数求导是纯数第一年和第二年都出现的核心技能。本文详细讲解例题 6.4.1，这是一道典型题，要求对包含 sin x 和 cos x 的函数求导、在给定点计算导数值，并求出切线方程。我们将逐步解释，让你能对考试题应用相同方法。
1. The AQA Context | AQA 考点背景
n the AQA specification, the derivatives of sin x and cos x are introduced under the topic 'Differentiation' ere x is measured in radians. These results are often tested alongside the chain rule, product rule, and quotient rule.
在 AQA 考纲中，sin x 和 cos x 的导数在“微分”主题下引入。要求掌握 d/dx (sin x) = cos x 以及 d/dx (cos x) = −sin x，其中 x 以弧度为单位。这些结果常与链式法则、乘法法则和除法法则一起考查。
d/dx (sin x) = cos x, d/dx (cos x) = −sin x
You must use radians for all calculus work unless the question explicitly says degrees. This is because the standard limits and derivative formulas for trigonometric functions are derived from radian measure.
除非题目明确说明使用角度制，所有微积分运算都必须使用弧度制。这是因为三角函数的标准极限和导数公式都是从弧度制推导出来的。
2. Key Standard Derivatives | 关键标准导数
Before solving Example 6.4.1, recall the two standard results. If f(x) = sin x, then f'(x) = cos x. If f(x) = cos x, then f' vative of cos x is a very common source of sign errors, so it must be memorised precisely.
在解例题 6.4.1 之前，先回顾两个标准结果。若 f(x) = sin x，则 f'(x) = cos x。若 f(x) = cos x，则 f'(x) = −sin x。cos x 导数中的负号是符号错误的常见来源，因此必须准确记忆。
sin x. The constant remains unchanged, while the trigonometric function differentiates as usual.
由于微分具有线性性质，常数相乘保留，和逐项求导。例如 d/dx (3 sin x) = 3 cos x，d/dx (−2 cos x) = 2 sin x。常数保持不变，三角函数照常求导。
3. Statement of Example 6.4.1 | 例题 6.4.1 陈述
formally. Example 6.4.1: The function f is defined by f(x) = 3 sin x − 2 cos x. (a) Find f'(x). (b) Evaluate f'4). (c) Find the equation of the tangent to the curve y = f(x) at the point where x = π/4.
现在正式陈述例题。例题 6.4.1：函数 f 定义为 f(x) = 3 sin x − 2 cos x。(a) 求 f'(x)。(b) 计算 f'(π/4)。(c) 求曲线 y = f(x) 在 x = π/4 处的切线方程。
This example is typical of AQA Paper 1 and Paper 2, where a multi-part question builds from differentiation to evaluation to tangent equation. The function uses both sin x and cos x so that the negative sign in the derivative of cos x is tested.
这道例题是 AQA 试卷一和试卷二的典型题，多小问从求导逐步过渡到求值和切线方程。函数同时包含 sin x 和 cos x，因此考查了 cos x 导数中的负号。
4. Differentiating Term by Term | 逐项求导
To find f' cos x. The derivative of 3 sin x is 3 cos x. The derivative of −2 cos x is −2 × (−sin x) =
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