Derivatives of Inverse Trigonometric Functions | 反三角函数的导数

📚 Derivatives of Inverse Trigonometric Functions | 反三角函数的导数

In IB Mathematics, differentiating inverse trigonometric functions is a fundamental skill that appears in calculus problems involving rates of change, optimisation, and integration. These functions—arcsin, arccos, arctan, and their reciprocal counterparts—are not just abstract inverses; they model angles whose trigonometric ratios are known. Mastering their derivatives requires a solid understanding of implicit differentiation, the chain rule, and careful attention to the principal values that keep the functions well‑defined. This article provides a complete guide, from formal definitions through step‑by‑step derivations to worked examples, focusing on the techniques and common pitfalls you will encounter in Analysis & Approaches HL and Applications & Interpretation HL.

在IB数学中，对反三角函数求导是一项基本技能，经常出现在涉及变化率、优化和积分的微积分问题中。这些函数——arcsin、arccos、arctan以及它们的倒数形式——不仅仅是抽象的逆；它们描述了已知三角比时所对应的角。掌握它们的导数需要扎实理解隐函数求导、链式法则，并特别注意保持函数良定义的主值区间。本文提供完整的指南，从正式定义到逐步推导，再到典型例题，聚焦于你在分析与方法HL和应用与解释HL中会遇到的方法和常见误区。

1. Introduction to Inverse Trigonometric Functions | 反三角函数简介

An inverse trigonometric function answers the question: “What angle has a given sine, cosine, or tangent?” Because the basic trigonometric functions are periodic, they are not one‑to‑one over their entire domains. To define inverses, we restrict each trigonometric function to a principal domain where it is monotonic. The resulting inverse functions are conventionally written as arcsin x (or sin⁻¹x), arccos x, arctan x, arccot x, arcsec x, and arccsc x. They return an angle measured in radians, and their derivatives exist everywhere except at certain endpoints or where the denominator becomes zero.

反三角函数回答这样一个问题：“哪个角的正弦、余弦或正切等于给定的值？”因为基本三角函数是周期性的，它们在整个定义域上不是单射的。为了定义反函数，我们把每个三角函数限制在它单调的主值区间上。由此得到的反函数习惯上写作 arcsin x（或 sin⁻¹x）、arccos x、arctan x、arccot x、arcsec x 和 arccsc x。它们返回一个以弧度为单位的角度，其导数在所有点都存在，除非在若干端点或分母为零处。

2. Review of Definitions and Principal Values | 反三角函数定义与主值回顾

Precise knowledge of the principal value ranges is essential when simplifying derivatives, because it determines the sign of trigonometric expressions like cos(arcsin x). The standard conventions are:

精确了解主值范围对于化简导数至关重要，因为它决定了诸如 cos(arcsin x) 等三角表达式的符号。标准约定如下：

arcsin x is defined for x ∈ [−1, 1], with range [−π/2, π/2]. In this interval, cos(arcsin x) ≥ 0.
arccos x is defined for x ∈ [−1, 1], with range [0, π]. In this interval, sin(arccos x) ≥ 0.
arctan x is defined for all real x, with range (−π/2, π/2). This ensures sec²(arctan x) > 0.
arccot x has range (0, π), arcsec x has range [0, π/2) ∪ (π/2, π], and arccsc x has range [−π/2, 0) ∪ (0, π/2].

arcsin x 的定义域为 x ∈ [−1, 1]，值域为 [−π/2, π/2]。在此区间内，cos(arcsin x) ≥ 0。
arccos x 的定义域为 x ∈ [−1, 1]，值域为 [0, π]。在此区间内，sin(arccos x) ≥ 0。
arctan x 的定义域为全体实数，值域为 (−π/2, π/2)。这确保 sec²(arctan x) > 0。
arccot x 的值域为 (0, π)，arcsec x 的值域为 [0, π/2) ∪ (π/2, π]，而 arccsc x 的值域为 [−π/2, 0) ∪ (0, π/2]。

These principal values guarantee that the derivatives we derive are single‑valued and consistent with the fundamental identities used in implicit differentiation.

这些主值保证了我们推导出的导数是单值的，并与隐函数求导中使用的基本恒等式一致。

3. The Method of Implicit Differentiation | 隐函数求导法

he derivative of an inverse trigonometric function is to rewrite the relationship in a non‑ trigonometric function of the inverse. For example, if y = arcsin x, then sin y = x and y ∈ rean identity, carefully choosing the sign dictated by the principal value range of y. This technique works for all six inverse trigonometric functions and reinforces your understanding of implicit differentiation and trigonometric identities.

求反三角函数导数最有效的方法是：将关系式写为非反函数形式，对其隐函数求导，然后消去反函数的三角函数。例如，若 y = arcsin x，则 sin y = x 且 y ∈ [−π/2, π/2]。两边同时对 x 求导得 cos y · dy/dx = 1，因此 dy/dx = 1/cos y。接着我们用勾股恒等式将 cos y 用 x 表示，根据 y 的主值范围仔细选取符号。这个方法适用于所有六种反三角函数，并能加深你对隐函数求导和三角恒等式的理解。

4. Derivative of arcsin x | 反正弦函数的导数

sin x. Then sin y = x, where −π/2 ≤ y ≤ π/2. Differentiating implicitly: cos y · dy/dx = 1 ⇒ dy/dx = 1/cos y. Using the identity cos² y = 1 − sin² y, we have cos y = ±√1 − x². Because y ∈ [−π/2, π/2], cos y is non‑negative. Hence cos y = √1 − x². Substituting back gives the standard formula:

令 y = arcsin x。则 sin y = x，其中 −π/2 ≤ y ≤ π/2。隐函数求导：cos y · dy/dx = 1 ⇒ dy/dx = 1/cos y。利用恒等式 cos² y = 1 − sin² y，我们有 cos y = ±√1 − x²。由于 y ∈ [−π/2, π/2]，cos y 非负。因此 cos y = √1 − x²。代回即得标准公式：

d/dx (arcsin x) = 1 / √1 − x², for |x| < 1.

cal. This formula is one of the most frequently used results in calculus, and remembering the denominator’s square root without a negative sign is essential.

注意导数在 x = ±1 处未定义，该处切线为垂直。该公式是微积分中最常用的结果之一，牢记分母中的平方根不带负号至关重要。

5. Derivative of arccos x | 反余弦函数的导数

s x, we have cos y = x with 0 ≤ y ≤ π. Implicit differentiation yields: −sin y · dy/dx = 1 ⇒ dy/dx = −1/sin y. In the principal range, sin y ≥ 0, so sin y = √1 − cos² y = √1 − x². Hence:

对于 y = arccos x，有 cos y = x，其中 0 ≤ y ≤ π。隐函数求导得：−sin y · dy/dx = 1 ⇒ dy/dx = −1/sin y。在主值区间内 sin y ≥ 0，故 sin y = √1 − cos² y = √1 − x²。于是：

d/dx (arccos x) = −1 / √1 − x², for |x| < 1.

π/2, a constant, so their derivatives must sum to zero. This relationship can serve as a quick check of your work.

注意到 arcsin x 和 arccos x 的导数互为相反数。这很合理，因为 arcsin x + arccos x = π/2 是常数，所以它们的导数之和必须为零。这个关系可以作为检查你计算是否正确的一种快捷方式。

6. Derivative of arctan x | 反正切函数的导数

= arctan x. Then tan y = x, with −π/2 < y < π/2. Differentiating: sec² y · dy/dx = 1 ⇒ dy/dx = 1/(1 + x²). No square root appears, and the derivative is defined for all real x.

令 y = arctan x。则 tan y = x，其中 −π/2 < y < π/2。求导：sec² y · dy/dx = 1 ⇒ dy/dx = 1/sec² y = cos² y。利用恒等式 sec² y = 1 + tan² y = 1 + x² 重写 sec² y。因为 cos² y = 1/sec² y，我们得到 dy/dx = 1/(1 + x²)。该导数中没有根号，且对全体实数 x 都有定义。

d/dx (arctan x) = 1 / (1 + x²), x ∈ ℝ.

ce of domain restrictions (other than all reals) makes arctan particularly manageable in calculus.

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这个简洁的有理函数在积分中极其有用，因为它直接给出原函数 ∫ 1/(1+x²) dx = arctan x + C。除了定义域为全体实数外没有其他限制，这使 arctan 在微积分中特别容易处理。

7. Derivatives of Other Inverse Trig Functions (arccot, arcsec, arccsc) | 其他反三角函数的导数

The remaining three inverse trigonometric functions have derivatives that can be derived similarly, though absolute values often appear due to sign considerations in the principal ranges. It is common to express them in a form that emphasises their close connection to the three basic derivatives.

其余三个反三角函数的导数可以类似推导，但由于主值区间中的符号考量，公式中常会出现绝对值。通常我们会以强调它们与三个基本导数紧密联系的形式来表达。

arccot x, we use tan(π/2 − y) = x or differentiate cot y = x directly. The derivative is:

对于 y = arccot x，可以利用 tan(π/2 − y) = x 或直接对 cot y = x 求导。其导数为：

d/dx (arccot x) = −1 / (1 + x²), x ∈ ℝ.

For y = arcsec x, the relation sec y = x (with y ∈ [0, π/2) ∪ (π/2, π]) gives:

对于 y = arcsec x，关系式 sec y = x（y ∈ [0, π/2) ∪ (π/2, π]）给出：

d/dx (arcsec x) = 1 / (|x| √x² − 1), |x| > 1.

For y = arccsc x, with csc y = x (y ∈ [−π/2, 0) ∪ (0, π/2]), the analogous process yields:

对于 y = arccsc x，csc y = x（y ∈ [−π/2, 0) ∪ (0, π/2]），类似的过程得到：

d/dx (arccsc x) = −1 / (|x| √x² − 1), |x| > 1.

The absolute value |x| is necessary because the square root √x²−1 is always positive, but the sign of tan y or cot y depends on the quadrant determined by x, and the absolute value consolidates the piecewise expression into one formula. If you cannot recall the absolute value, you can always re‑derive it by considering the sign of the relevant trigonometric function in the principal range.

绝对值 |x| 是必要的，因为平方根 √x²−1 总是正的，而 tan y 或 cot y 的符号取决于 x 所决定的象限，绝对值将分段表达式统一为一个公式。如果你记不住绝对值，可以始终根据主值区间中相应三角函数的符号重新推导。

8. Chain Rule with Inverse Trig Functions | 反三角函数的链式法则应用

standard derivative (with argument u) by du/dx. For example, if f(x) = arcsin(5x²), then f'(x) = [1/√(1−(5x²)²)] · d/dx(5x²) = (1/√1−25x⁴) · 10x. Similarly, for arctan(eˣ) the derivative is [1/(1 + (eˣ)²)] · eˣ = eˣ/(1 + e²ˣ).

在大多数 IB 问题中，你并不是对单独的 arcsin x 求导，而是对像 arcsin(u(x)) 这样的复合函数求导。链式法则将每个基本导数推广为：将标准导数（以 u 为自变量）乘以 du/dx。例如，若 f(x) = arcsin(5x²)，则 f'(x) = [1/√(1−(5x²)²)] · d/dx(5x²) = (1/√1−25x⁴) · 10x。类似地，对于 arctan(eˣ)，导数为 [1/(1 + (eˣ)²)] · eˣ = eˣ/(1 + e²ˣ)。

the absolute value of the inner function must be handled correctly: d/dx arcsec(u) = [1/(|u| √u²−1)] · du/dx. Failing to include |u| often loses marks in HL papers.

当 u(x) 本身在反三角函数的参数中含有根号或平方时，要特别注意，因为你需要仔细化简平方根下的表达式。对于 arcsec 和 arccsc，必须正确处理内部函数的绝对值：d/dx arcsec(u) = [1/(|u| √u²−1)] · du/dx。在 HL 试卷中忘记写 |u| 经常会丢分。

Another use of the chain rule arises when inverse trigonometric functions appear as part of a product or quotient. For instance, differentiating x² arccos(3x) requires the product rule, where the derivative of arccos(3x) involves the chain rule. Treat these as layered applications of the differentiation toolkit you have already built.

当反三角函数出现在乘积或分式中时，链式法则也会再次出现。例如，对 x² arccos(3x) 求导需要乘法法则，其中 arccos(3x) 的导数又涉及链式法则。将这些情况看作是对你已经建立的微分工具箱的分层应用。

9. Common Pitfalls and Domain Issues | 常见错误与定义域问题

r is forgetting the negative sign in the derivative of arccos x and arccot x. Another is misapplying the square root: always ensure you are taking the positive root based on the principal range, and keep in mind that the expression under the root must be non‑ erivative is undefined at the endpoints x = ±1, so stating the domain of the derivative as |x| < 1 is important.

在反三角函数导数的作业中，有几个错误反复出现。意识到这些错误会为你保住宝贵的分数。一个经典错误是忘记 arccos x 和 arccot x 导数中的负号。另一个是错误地处理平方根：永远要根据主值区间确保取正根，并记住根号下的表达式必须非负。对于 arcsin 和 arccos，在端点 x = ±1 处导数未定义，因此将导数的定义域声明为 |x| < 1 是很重要的。

g nature on (1, ∞) and decreasing on (−∞, −1). The absolute value handles this automatically. As a sanity check, you can always graph the original function and its derivative on your GDC to verify the slope’s sign.

对于 arcsec 和 arccsc，最常见的错误是遗漏 x 或 u(x) 外面的绝对值符号。如果没有绝对值，对于负的 x 导数就会给出错误的符号。你可以快速验证一下：arcsec(−2) = 2π/3，考虑到函数在 (1, ∞) 上递增而在 (−∞, −1) 上递减，其导数应为负。绝对值能自动处理这一点。作为一种合理性检验，你随时都可以在图形计算器（GDC）上画出原函数及其导数的图像，以验证斜率的符号。

10. Worked Examples | 例题解析

Example 1: Differentiate f(x) = arcsin(2x − 1). Find the domain where f'(x) is defined.

例题 1：求 f(x) = arcsin(2x − 1) 的导数。找出 f'(x) 有定义的定义域。

Let u = 2x − 1. Then f'(x) = [1/√1 − u²] · u' = 2 / √(1 − (2x−1)²). The derivative exists when the denominator is non‑ olve −1 < 2x−1 < 1 → 0 < x < 1. Inside this interval, 1 − (2x−1)² > 0, so f'(x) is defined for 0 < x < 1.

令 u = 2x − 1。则 f'(x) = [1/√1 − u²] · u' = 2 / √(1 − (2x−1)²)。当分母不为零且 arcsin 的参数在 (−1, 1) 内时导数存在。解 −1 < 2x−1 < 1 → 0 < x < 1。在此区间内，1 − (2x−1)² > 0，所以 f'(x) 在 0 < x < 1 上有定义。

Example 2: Given y = arctan(ln x), find dy/dx and evaluate it at x = e.

例题 2：已知 y = arctan(ln x)，求 dy/dx 并在 x = e 处求值。

· (1/x) = 1 / [x (1 + (ln x)²)]. At x = e, ln e = 1, so dy/dx = 1 / [e (1 + 1²)] = 1/(2e).

运用链式法则：dy/dx = [1/(1 + (ln x)²)] · (1/x) = 1 / [x (1 + (ln x)²)]。在 x = e 处，ln e = 1，所以 dy/dx = 1 / [e (1 + 1²)] = 1/(2e)。

Example 3: Differentiate g(t) = t² arcsec(t/2). State any restrictions on t.

例题 3：对 g(t) = t² arcsec(t/2) 求导。指出 t 的任何限制。

This requires the product rule: g'(t) = 2t · arcsec(t/2) + t² · [d/dt arcsec(t/2)]. For the inner derivative, let u = t/2

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