📚 The Derivatives of Standard Inverse Trigonometrical Functions | 标准反三角函数的导数
In AQA A-Level Mathematics, inverse trigonometric derivatives appear in pure mathematics topics such as differentiating inverse functions, integration leading to inverse trig forms, and solving differential equations. They are not isolated results; examiners often mix them with the chain rule, product rule and implicit differentiation. This article covers the standard derivatives, their derivations from implicit differentiation, common errors and exam-style applications.
在 AQA A-Level 数学中，反三角函数导数出现在纯数学主题中，例如反函数求导、导出反三角形式的积分以及求解微分方程。它们不是孤立的结果，考官常常将它们与链式法则、乘积法则和隐函数求导混合考查。本文介绍标准导数、用隐函数求导推导公式、常见错误以及考试风格的应用。
1. Why These Derivatives Matter | 为什么这些导数重要
The inverse trigonometric functions arcsin x, arccos x and arctan x are needed whenever an angle is expressed as a function of a ratio. Their derivatives are essential for differentiating inverse functions, for integrating expressions such as 1/√1 - x² and 1/(1 + x²), and for solving related rates or differential equations problems. AQA questions frequently require you to quote these results accurately and then use the chain rule.
当角度被表示为一个比值函数时，就需要用到反三角函数 arcsin x、arccos x 和 arctan x。它们的导数对于反函数求导、积分形如 1/√1 - x² 和 1/(1 + x²) 的表达式，以及解决相关变化率或微分方程问题都至关重要。AQA 试题经常要求你准确引用这些结果，然后使用链式法则。
You can derive all three core results from implicit differentiation, so memorising the derivation also helps you check signs and conditions if you forget a formula in the exam.
你可以通过隐函数求导推导出所有三个核心结果，因此记住推导过程也有助于在考试中忘记公式时检查符号和条件。
2. Standard Results at a Glance | 标准结果速览
The three core derivatives required for AQA are shown below. For arcsin x and arccos x, the gradient is real and finite only when |x| < 1. For arctan x, the derivative is defined for all real x.
AQA 要求的三个核心导数如下所示。对于 arcsin x 和 arccos x，仅当 |x| < 1 时梯度才是实数且有限。对于 arctan x，导数对所有实数 x 都有定义。
| Function | Derivative | Valid domain for finite gradient |
|---|---|---|
| y = arcsin x | dy/dx = 1/√1 - x² | |x| < 1 |
| y = arccos x | dy/dx = -1/√1 - x² | |x| < 1 |
| y = arctan x | dy/dx = 1/(1 + x²) | all real x |
Some texts write arcsin x as sin⁻¹ x. Do not confuse sin⁻¹ x with (sin x)⁻¹ = 1/sin x.
有些教材将 arcsin x 写作 sin⁻¹ x。不要将 sin⁻¹ x 与 (sin x)⁻¹ = 1/sin x 混淆。
3. Derivative of arcsin x | arcsin x 的导数
th respect to x: 1 = cos y dy/dx, so dy/dx = 1/cos y. On this range, cos y ≥ 0, so cos y = √1 - sin² y = √1 - x². Therefore the derivative is positive.
设 y = arcsin x。则 x = sin y，其主值范围为 -π/2 ≤ y ≤ π/2。对 x = sin y 关于 x 隐函数求导：1 = cos y dy/dx，所以 dy/dx = 1/cos y。在此范围内，cos y ≥ 0，因此 cos y = √1 - sin² y = √1 - x²。所以导数为正。
d/dx (arcsin x) = 1/√1 - x²
the gradient becomes infinite at the endpoints and is not defined there as a finite value.
y = arcsin x 的图像在 x = -1 和 x = 1 处具有竖直切线，因此在端点处梯度趋于无穷，不能作为有限值定义。
4. Derivative of arccos x | arccos x 的导数
implicitly: 1 = -sin y dy/dx, so dy/dx = -1/sin y. Since sin y ≥ 0 on this range, sin y = √1 - cos² y = √1 - x². The negative sign appears because cos y is decreasing on 0 ≤ y ≤ π.
设 y = arccos x。则 x = cos y，其主值范围为 0 ≤ y ≤ π。隐函数求导：1 = -sin y dy/dx，所以 dy/dx = -1/sin y。因为在此范围内 sin y ≥ 0，所以 sin y = √1 - cos² y = √1 - x²。负号的出现是因为 cos y 在 0 ≤ y ≤ π 上单调递减。
d/dx (arccos x) = -1/√1 - x²
n x + arccos x = π/2, their derivatives must sum to zero, which gives another quick check.
请注意，arcsin x 和 arccos x 的导数只差一个符号。因为 arcsin x + arccos x = π/2，它们的导数相加必须为零，这也是一个快速检验方法。
5. Derivative of arctan x | arctan x 的导数
dy/dx, so dy/dx = 1/sec² y. Using sec² y = 1 + tan² y = 1 + x² gives the standard result.
设 y = arctan x。则 x = tan y，其主值范围为 -π/2 < y < π/2。隐函数求导：1 = sec² y dy/dx，所以 dy/dx = 1/sec² y。利用 sec² y = 1 + tan² y = 1 + x²，得到标准结果。
d/dx (arctan x) = 1/(1 + x²)
Unlike arcsin x and arccos x, the derivative of arctan x has no square root and no restricted real domain. This makes it particularly useful in integration and in problems involving rational expressions.
与 arcsin x 和 arccos x 不同，arctan x 的导数没有平方根，也没有实数定义域限制。这使得它在积分和涉及有理表达式的问题中特别有用。
6. Using the Chain Rule | 结合链式法则
In most exam questions the argument is not simply x. If u is a function of x, then the chain rule gives:
在大多数考试题中，变量不单是 x。如果 u 是 x 的函数，那么链式法则给出：
- d/dx (arcsin u) = 1/√1 - u² × du/dx
- d/dx (arccos u) = -1/√1 - u² × du/dx
- d/dx (arctan u) = 1/(1 + u²) × du/dx
For example, if y = arcsin(2x), then dy/dx = 1/√(1 - (2x)²) × 2 = 2/√1 - 4x². If y = arctan(3x²), then dy/dx = 1/(1 + (3x²)²) × 6x = 6x/(1 + 9x⁴).
例如，如果 y = arcsin(2x)，则 dy/dx = 1/√(1 - (2x)²) × 2 = 2/√1 - 4x²。如果 y = arctan(3x²)，则 dy/dx = 1/(1 + (3x²)²) × 6x = 6x/(1 + 9x⁴)。
Always identify the inner function u before differentiating, and remember to multiply by
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