The derivatives of standard inverse trigonometrical functions | 标准反三角函数的导数

📚 The derivatives of standard inverse trigonometrical functions | 标准反三角函数的导数

Inverse trigonometric functions are an essential part of A-Level Mathematics. Understanding their derivatives allows you to solve a wide range of differentiation problems, especially those involving composite functions and applications to rates of change.

反三角函数是 A-Level 数学中的重要组成部分。掌握它们的导数，可以帮助你解决大量微分问题，特别是涉及复合函数和变化率应用的问题。

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1. Why do we need inverse trigonometric derivatives? | 为什么需要反三角函数的导数？

When we differentiate expressions such as sin⁻¹ x, cos⁻¹ x or tan⁻¹ x, we cannot use the ordinary rules for sin x, cos x and tan x directly. The inverse functions have their own distinct derivative formulas.

当我们对 sin⁻¹ x、cos⁻¹ x 或 tan⁻¹ x 等表达式求导时，不能直接使用 sin x、cos x、tan x 的普通求导法则。反函数具有自己独特的导数公式。

These formulas are derived from the fact that if y = f⁻¹(x), then f(y) = x. By implicit differentiation, we can obtain a general rule for the derivative of an inverse function.

这些公式源于这样一个事实：如果 y = f⁻¹(x)，那么 f(y) = x。通过隐函数求导，我们可以得到反函数导数的一般法则。

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2. The general inverse function rule | 反函数求导的一般法则

Let y = f⁻¹(x). Then f(y) = x. Differentiating both sides with respect to x gives:

设 y = f⁻¹(x)，则 f(y) = x。两边对 x 求导得：

f′(y) · dy/dx = 1

Therefore:

因此：

dy/dx = 1 / f′(y)

This rule is the foundation for all the derivatives of inverse trigonometric functions. We apply it to sin, cos and tan to obtain their standard results.

这个法则是所有反三角函数导数的基础。我们将它应用于 sin、cos、tan，从而得到它们的标准结果。

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3. Derivative of sin⁻¹ x | sin⁻¹ x 的导数

Let y = sin⁻¹ x. Then sin y = x. Since x is restricted to the interval [−1, 1] and y is in [−π/2, π/2], we can safely use cos y = √(1 − x²).

设 y = sin⁻¹ x，则 sin y = x。由于 x 限制在区间 [−1, 1] 内，且 y 在 [−π/2, π/2] 内，我们可以安全地使用 cos y = √(1 − x²)。

Differentiating sin y = x implicitly gives:

对 sin y = x 隐函数求导得：

cos y · dy/dx = 1

Hence:

因此：

d/dx(sin⁻¹ x) = 1 / √(1 − x²)

This formula is valid for −1 < x < 1. At x = ±1, the denominator becomes zero and the derivative is undefined.

该公式在 −1 < x < 1 时成立。在 x = ±1 处，分母为零，导数不存在。

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4. Derivative of cos⁻¹ x | cos⁻¹ x 的导数

Let y = cos⁻¹ x. Then cos y = x, with y in [0, π]. Differentiating implicitly:

设 y = cos⁻¹ x，则 cos y = x，其中 y 在 [0, π] 内。隐函数求导：

−sin y · dy/dx = 1

Since sin y = √(1 − x²) for y in [0, π], we obtain:

因为在 [0, π] 内 sin y = √(1 − x²)，所以我们得到：

d/dx(cos⁻¹ x) = −1 / √(1 − x²)

Notice the negative sign. This is a common source of errors, so be careful when differentiating expressions that contain cos⁻¹ x.

注意这里的负号。这是常见的错误来源，所以对包含 cos⁻¹ x 的表达式求导时要格外小心。

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5. Derivative of tan⁻¹ x | tan⁻¹ x 的导数

Let y = tan⁻¹ x. Then tan y = x. Differentiating implicitly using sec²y:

设 y = tan⁻¹ x，则 tan y = x。利用 sec²y 进行隐函数求导：

sec²y · dy/dx = 1

Using the identity sec²y = 1 + tan²y = 1 + x², we get:

利用恒等式 sec²y = 1 + tan²y = 1 + x²，我们得到：

d/dx(tan⁻¹ x) = 1 / (1 + x²)

This formula is valid for all real values of x, making tan⁻¹ x particularly easy to work with in integration and differentiation.

该公式对所有实数 x 均成立，这使得 tan⁻¹ x 在积分和微分中都非常容易处理。

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6. Derivatives of sec⁻¹ x, csc⁻¹ x and cot⁻¹ x | sec⁻¹ x、csc⁻¹ x 和 cot⁻¹ x 的导数

These are less commonly tested at A-Level, but you may encounter them in extension questions. Their derivatives are:

这些在 A-Level 中不常考，但你可能在扩展题中遇到。它们的导数为：

• d/dx(sec⁻¹ x) = 1 / (|x| √(x² − 1)) for |x| > 1
• d/dx(sec⁻¹ x) = 1 / (|x| √(x² − 1))，其中 |x| > 1
• d/dx(csc⁻¹ x) = −1 / (|x| √(x² − 1)) for |x| > 1
• d/dx(csc⁻¹ x) = −1 / (|x| √(x² − 1))，其中 |x| > 1
• d/dx(cot⁻¹ x) = −1 / (1 + x²) for all real x
• d/dx(cot⁻¹ x) = −1 / (1 + x²)，对所有实数 x 成立

Notice that sec⁻¹ and csc⁻¹ require the absolute value |x| in the denominator. Different books may use slightly different definitions, so always follow the convention used in your exam board.

注意 sec⁻¹ 和 csc⁻¹ 的分母中需要绝对值 |x|。不同教材可能使用略有不同的定义，因此请始终遵循你考试局所采用的约定。

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7. Applying the chain rule | 链式法则的应用

In most exam questions, the argument of the inverse trigonometric function is not simply x, but a function of x such as 2x, x² or eˣ. We must combine the standard derivative with the chain rule.

在大多数考题中，反三角函数的自变量并不是简单的 x，而是 x 的函数，例如 2x、x² 或 eˣ。我们必须将标准导数与链式法则结合使用。

For example, if y = sin⁻¹(3x), then:

例如，若 y = sin⁻¹(3x)，则：

dy/dx = 3 / √(1 − (3x)²)

Similarly, if y = tan⁻¹(x²), then:

同样，若 y = tan⁻¹(x²)，则：

dy/dx = 2x / (1 + x⁴)

Always remember to multiply by the derivative of the inside function.

永远记得乘以内层函数的导数。

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8. Useful identities for simplifying | 化简时常用的恒等式

Some expressions simplify before differentiation. The following identities are helpful:

有些表达式在求导前可以化简。以下恒等式很有帮助：

• sin⁻¹ x + cos⁻¹ x = π/2
• sin⁻¹ x + cos⁻¹ x = π/2
• tan⁻¹ x + cot⁻¹ x = π/2
• tan⁻¹ x + cot⁻¹ x = π/2
• sec⁻¹ x + csc⁻¹ x = π/2 for x ≥ 1
• sec⁻¹ x + csc⁻¹ x = π/2，其中 x ≥ 1

Because the derivative of a constant is zero, these identities often show why the derivative of sin⁻¹ x is the negative of the derivative of cos⁻¹ x.

因为常数的导数为零，这些恒等式常常解释为什么 sin⁻¹ x 的导数是 cos⁻¹ x 的导数的相反数。

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9. Worked example 1 | 例题 1

Find the derivative of y = cos⁻¹(2x).

求 y = cos⁻¹(2x) 的导数。

Solution: Using the chain rule:

解法：使用链式法则：

dy/dx = −1 / √(1 − (2x)²) × 2

dy/dx = −2 / √(1 − 4x²)

The domain is |x| < 1/2, so the square root is defined.

定义域为 |x| < 1/2，因此根号有意义。

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10. Worked example 2 | 例题 2

Differentiate y = tan⁻¹(√x).

求 y = tan⁻¹(√x) 的导数。

Solution: Let u = √x = x^(1/2). Then:

解法：设 u = √x = x^(1/2)。则：

du/dx = 1 / (2√x)

Using the chain rule:

使用链式法则：

dy/dx = 1 / (1 + (√x)²) × 1 / (2√x)

dy/dx = 1 / (2√x (1 + x))

This type of question tests both the standard formula and your ability to handle fractional powers.

这类问题既考查标准公式，也考查你处理分数次幂的能力。

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11. Common mistakes to avoid | 需要避免的常见错误

Students often lose marks on inverse trigonometric derivatives for the following reasons:

学生经常因以下原因在反三角函数导数上失分：

• Forgetting the negative sign in d/dx(cos⁻¹ x) and d/dx(cot⁻¹ x).
• 忘记 d/dx(cos⁻¹ x) 和 d/dx(cot⁻¹ x) 中的负号。
• Confusing sin⁻¹ x with 1/sin x. The notation sin⁻¹ x means the inverse sine, not the reciprocal.
• 混淆 sin⁻¹ x 与 1/sin x。符号 sin⁻¹ x 表示反正弦，而不是倒数。
• Omitting the chain rule when the argument is not x.
• 当自变量不是 x 时忽略了链式法则。
• Misusing the domain: d/dx(sin⁻¹ x) is undefined for |x| > 1.
• 误用定义域：当 |x| > 1 时，d/dx(sin⁻¹ x) 无定义。

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12. Summary and practice tips | 总结与练习建议

Memorise these three core formulas first:

首先牢记这三个核心公式：

Function | 函数	Derivative | 导数
sin⁻¹ x	1 / √(1 − x²)
cos⁻¹ x	−1 / √(1 − x²)
tan⁻¹ x	1 / (1 + x²)

Practise applying the chain rule to these formulas using linear functions, polynomials and exponentials. The more comfortable you are with the pattern, the faster you will solve exam questions.

练习将链式法则应用于这些公式，使用线性函数、多项式和指数函数作为内层函数。你对这些模式越熟练，解题速度就越快。

Keep a list of the formulas visible while doing homework, but aim to recall them without notes before your exam. Good luck!

做作业时可以把公式表放在旁边，但考试前要争取不用笔记就能回忆出来。祝你好运！

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