📚 Differentiating Trigonometric Functions | 三角函数的微分
In A-Level Edexcel Mathematics, differentiating trigonometric functions is a vital skill that connects algebraic techniques with periodic phenomena. Once you know the derivatives of sin x, cos x and tan x, you can use the chain, product and quotient rules to tackle a wide range of problems involving compound angles and products of functions.
在 A-Level Edexcel 数学中，三角函数的微分是一项将代数技巧与周期现象联系起来的重要技能。一旦你掌握了 sin x、cos x 和 tan x 的导数，就可以运用链式法则、乘积法则和商法则来处理涉及复合角度和函数乘积的各种问题。
1. Derivative of sin x | sin x 的导数
The standard result for differentiating sin x is that its derivative is cos x. This holds true only when x is measured in radians – a condition always assumed in A-Level calculus.
对 sin x 求导的标准结果是其导数为 cos x。这个结论只在 x 以弧度为单位时才成立——这也是 A-Level 微积分中始终默认的前提。
d/dx (sin x) = cos x
You can interpret this geometrically: the gradient of the sine curve at any point equals the cosine value at that point. The result is fundamental and must be memorised accurately.
你可以从几何角度理解：正弦曲线上任意一点的斜率等于该点的余弦值。这个结果是基础性的，必须准确记住。
he constant is preserved: d/dx (k sin x) = k cos x. For example, d/dx (5 sin x) = 5 cos x.
当对 sin x 与常数的乘积求导时，常数保留：d/dx (k sin x) = k cos x。例如，d/dx (5 sin x) = 5 cos x。
2. Derivative of cos x | cos x 的导数
The derivative of cos x with respect to x is −sin x. The negative sign is a common source of error, so always double‑check it.
cos x 对 x 的导数是 −sin x。负号是一个常见的错误来源，因此务必仔细核对。
d/dx (cos x) = −sin x
ations. For instance, d/dx (4 cos x) = −4 sin x, and d/dx (sin x + cos x) = cos x − sin x.
结合 sin x 的导数，你可以求出许多简单组合的导数。例如，d/dx (4 cos x) = −4 sin x，以及 d/dx (sin x + cos x) = cos x − sin x。
It is useful to remember that differentiating cos x shifts the graph and changes the sign, whereas sin x simply becomes cos x without a sign change.
记住，求导 cos x 会使图像发生平移并改变符号，而 sin x 则直接变为 cos x，符号不变，这很有帮助。
3. Derivative of tan x | tan x 的导数
The derivative of tan x is sec² x. Although you can derive it using the quotient rule with sin x and cos x, the result is worth memorising.
tan x 的导数是 sec² x。虽然你可以用 sin x 和 cos x 通过商法则推导出来，但这个结果仍然值得记住。
d/dx (tan x) = sec² x
e. This matches the shape of the tan curve, which always increases between its vertical asymptotes.
由于 sec² x 总是正值，y = tan x 的斜率永远不会为负。这与 tan 曲线在其垂直渐近线之间总是上升的形状相符。
In questions, you may need to differentiate tan(kx) or combine it with other rules. Always begin by writing tan x in terms of sin and cos if you need to prove the derivative.
在考试题目中，你可能需要对 tan(kx) 求导，或将其与其他法则结合。如果需要证明导数，始终从将 tan x 写成 sin 和 cos 的形式开始。
4. Using the Chain Rule with Trigonometric Functions | 链式法则与三角函数的运用
When the argument of a trigonometric function is a linear function of x, such as ax + b, the chain rule gives an extra factor equal to the coefficient of x.
当三角函数的参数是 x 的线性函数，例如 ax + b 时，链式法则会给出一个等于 x 系数的额外因子。
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)
(3x + 2). The general rule is to differentiate the outer function and multiply by the derivative of the inner function.
例如，d/dx (sin(3x + 2)) = 3 cos(3x + 2)。一般规则是：对外层函数求导，再乘以内层函数的导数。
When the inner function is more complicated, such as sin(x²) or cos(√x iply by the derivative of the inner expression. For instance, d/dx [sin(x²)] = 2x cos(x²).
当内层函数更复杂时，例如 sin(x²) 或 cos(√x)，你仍然需要乘以内层表达式的导数。例如，d/dx [sin(x²)] = 2x cos(x²)。
5. Product Rule with Trigonometric Functions | 乘积法则与三角函数
Many exam problems involve products such as x·sin x or eˣ·cos x. The product rule states that if y = uv, then dy/dx = u'v + uv'.
许多考试题涉及诸如 x·sin x 或 eˣ·cos x 的乘积。乘积法则指出，若 y = uv，则 dy/dx = u'v + uv'。
For y = x² sin x, let u = x² (u' = 2x) and v = sin x (v' en dy/dx = 2x sin x + x² cos x. Always keep the order consistent when writing your answer.
对于 y = x² sin x，令 u = x² (u' = 2x)，v = sin x (v' = cos x)。那么 dy/dx = 2x sin x + x² cos x。书写答案时务必保持顺序一致。
os x, apply the product rule in stages or learn the extended product rule. However, most A‑Level questions will involve only two factors.
遇到三个因子时，如 y = x·sin x·cos x，可分阶段应用乘积法则，或学习扩展的乘积法则。不过大多数 A‑Level 题目只涉及两个因子。
6. Quotient Rule with Trigonometric Functions | 商法则与三角函数
quotient rule is used when one function is divided by another: if y = u/v, then dy/dx = (u'v − uv') / v². It is essential for differentiating expressions like (sin x)/(1 + cos x).
当一个函数除以另一个函数时，需使用商法则：若 y = u/v，则 dy/dx = (u'v − uv') / v²。这对于 (sin x)/(1 + cos x) 这类表达式的求导至关重要。
x)) / cos²x = (cos²x + sin²x) / cos²x = 1 / cos²x = sec²x, confirming the earlier result.
为了对 y = sin x / cos x（即 tan x）求导，设 u = sin x，v = cos x。那么 /dx = (cos x·cos x − sin x·(−sin x)) / cos²x = (cos²x + sin²x) / cos²x = 1 / cos²x = sec²x，这验证了之前的结果。
Be careful with signs in the numerator: subtracting uv' often causes mistakes, especially when v' contains a negative sign. A systematic layout minimises errors.
留意分子中的符号：减去 uv' 时容易出错，特别是当 v' 包含负号时。有条理的书写方式能减少错误。
7. Derivatives of sec x, csc x and cot x | sec x、csc x 和 cot x 的导数
Although the derivatives of secant, cosecant and cotangent are not always given on the formula sheet, you can find them by rewriting the functions in terms of sin and cos and applying the chain or quotient rule.
虽然 secant、cosecant 和 cotangent 的导数并不总是出现在公式表中，但你可以通过将它们改写成 sin 和 cos 的形式，并应用链式法则或商法则来求得。
d/dx (sec x) = sec x tan x
d/dx (csc x) = −csc x cot x
d/dx (cot x) = −csc² x
s²x = sec x tan x. Memorising these can save time, but you should know how to derive them.
对于 sec x = 1/cos x，使用链式法则：d/dx (cos x)⁻¹ = −1·(cos x)⁻²·(−sin x) = sin x / cos²x = sec x tan x。记住这些结果可以节省时间，但你也应清楚推导过程。
8. Higher Order Derivatives of Trigonometric Functions | 三角函数的更高阶导数
You may be asked to find the second derivative, d²y/dx², of a trigonometric function, particularly when determining stationary points or nature of turning points.
你可能会被要求求三角函数的二阶导数 d²y/dx²，尤其是在判断驻点或拐点性质时。
derivative is −sin x, second derivative is −cos x. Notice that differentiating twice brings you back to a multiple of the original function.
从 y = sin x 开始，dy/dx = cos x，d²y/dx² = −sin x。对于 y = cos x，一阶导数是 −sin x，二阶导数是 −cos x。注意到求导两次会回到原函数的某个倍数。
h can be obtained by differentiating sec² x using the chain rule. Practice higher derivatives with chain and product rules to build fluency.
对于 tan x，d²y/dx² = 2 sec² x tan x，这可以通过对 sec² x 使用链式法则得到。通过练习包含链式法则和乘积法则的高阶导数，可以提升熟练度。
9. Applying Trigonometric Differentiation: Tangents, Normals and Rates of Change | 三角函数微分的应用：切线、法线与变化率
Once you have the derivative, you can find the equation of the tangent or normal to a trigonometric curve at a given point. This is a common A‑Level problem.
一旦求得导数，你就可以求出三角函数曲线在某点的切线或法线方程。这是 A‑Level 中常见的问题。
For y = sin x at x = π/4, dy/dx = cos(π/4) = √2/2. The gradient of the normal is then −√2. Use y − y₁ = m(x − x₁) to form the linear equation.
对于 y = sin x 在 x = π/4 处，dy/dx = cos(π/4) = √2/2。法线的斜率则为 −√2。使用 y − y₁ = m(x − x₁) 写出直线方程。
Connected rates of change problems may involve trigonometric expressions where one variable changes with respect to time, requiring implicit differentiation with chain rule.
相关变化率问题可能涉及三角函数，其中一个变量随时间变化，这需要使用链式法则进行隐式微分。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most frequent errors is forgetting the negative sign when differentiating cos x. Write the formula clearly at the top of your working to avoid this.
最常见的错误之一是在对 cos x 求导时忘记负号。在解题步骤的开头清晰写出公式可以避免这一点。
Another pitfall is mixing up radians and degrees: calculus with trigonometric functions only works in radians. Always check your calculator mode before evaluating a derivative numerically.
另一个陷阱是混淆弧度和角度：三角函数的微积分只能在弧度下进行。在数值计算导数之前，务必检查计算器模式。
When using the chain rule, be methodical: identify the outer function, differentiate it, then multiply by the derivative of the inner function. Missing the inner derivative accounts for many lost marks.
使用链式法则时应有条不紊：识别外层函数，对其求导，再乘以内层函数的导数。遗漏内层导数导致了许多失分。
Finally, simplify your answers where possible. Examiners expect final expressions such as (x cos x − sin x)/x² rather than leaving partially simplified fractions.
最后，尽可能化简答案。考官期望诸如 (x cos x − sin x)/x² 这样的最终表达式，而不是留下部分化简的分数。
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