Differentiation of Hyperbolic Functions | 双曲函数的微分

📚 Differentiation of Hyperbolic Functions | 双曲函数的微分

Hyperbolic functions appear frequently in advanced mathematics, physics and engineering. This guide covers their definitions, differentiation rules and common exam techniques for AQA A-Level Mathematics.

双曲函数在高等数学、物理学和工程学中频繁出现。本指南涵盖它们的定义、微分法则以及AQA A-Level数学中的常见考试技巧。

1. Definitions of Hyperbolic Functions | 双曲函数的定义

The hyperbolic sine, cosine and tangent functions are defined using exponential functions:

双曲正弦、余弦和正切函数是通过指数函数定义的：

sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x

Three further functions are the reciprocals: sech x = 1/cosh x, csch x = 1/sinh x and coth x = 1/tanh x.

另外三个函数是倒数：sech x = 1/cosh x，csch x = 1/sinh x，coth x = 1/tanh x。

The identity cosh² x − sinh² x = 1 is analogous to the Pythagorean identity for trigonometric functions, and is essential for simplifying derivatives.

恒等式 cosh² x − sinh² x = 1 与三角函数的勾股恒等式类似，在化简导数时至关重要。

2. Basic Derivatives of Hyperbolic Functions | 双曲函数的基本导数

The derivatives of sinh x and cosh x are directly obtained from their exponential definitions:

sinh x 和 cosh x 的导数可以直接从它们的指数定义得到：

d/dx (sinh x) = cosh x, d/dx (cosh x) = sinh x

Notice that unlike trigonometric functions, there is no sign change for cosh x. This is because cosh x = (eˣ + e⁻ˣ)/2, whose derivative is (eˣ − e⁻ˣ)/2 = sinh x.

注意与三角函数不同，cosh x 的导数没有符号变化。这是因为 cosh x = (eˣ + e⁻ˣ)/2，其导数为 (eˣ − e⁻ˣ)/2 = sinh x。

For tanh x, using the quotient rule or the definition tanh x = sinh x/cosh x gives:

对于 tanh x，使用商法则或定义 tanh x = sinh x/cosh x 可得：

d/dx (tanh x) = sech² x

Similarly:

类似地：

d/dx (coth x) = −csch² x

d/dx (coth x) = −csch² x
d/dx (sech x) = −sech x tanh x

d/dx (sech x) = −sech x tanh x
d/dx (csch x) = −csch x coth x

d/dx (csch x) = −csch x coth x

Memorising these six derivatives is essential for solving problems quickly in the exam.

牢记这六个导数是在考试中快速解题的关键。

3. Derivative of sinh x from First Principles | 从第一性原理推导 sinh x 的导数

Using the definition sinh x = (eˣ − e⁻ˣ)/2 and differentiating term by term:

利用定义 sinh x = (eˣ − e⁻ˣ)/2 并逐项微分：

d/dx (sinh x) = (eˣ + e⁻ˣ)/2 = cosh x

Similarly, d/dx (cosh x) = (eˣ − e⁻ˣ)/2 = sinh x. This exponential approach confirms the derivative formulas without needing to memorise them separately.

类似地，d/dx (cosh x) = (eˣ − e⁻ˣ)/2 = sinh x。这种指数方法可以确认导数公式，而无需单独记忆。

Since tanh x = sinh x/cosh x, applying the quotient rule yields:

由于 tanh x = sinh x/cosh x，应用商法则可得：

d/dx (tanh x) = (cosh x · cosh x − sinh x · sinh x) / cosh² x = 1/cosh² x = sech² x

4. Chain Rule with Hyperbolic Functions | 双曲函数的链式法则

When the argument is not simply x, the chain rule is required. For example, if y = sinh(3x), then dy/dx = 3 cosh(3x).

当自变量不是简单的 x 时，需要使用链式法则。例如，若 y = sinh(3x)，则 dy/dx = 3 cosh(3x)。

In general, for a function u = f(x):

一般地，对于函数 u = f(x)：

d/dx (sinh u) = cosh u · du/dx

d/dx (cosh u) = sinh u · du/dx

d/dx (tanh u) = sech² u · du/dx

Example: Differentiate y = cosh(x² + 1). Let u = x² + 1, then du/dx = 2x, so dy/dx = 2x sinh(x² + 1).

例：求 y = cosh(x² + 1) 的导数。令 u = x² + 1，则 du/dx = 2x，所以 dy/dx = 2x sinh(x² + 1)。

5. Product Rule with Hyperbolic Functions | 双曲函数的乘积法则

When a hyperbolic function is multiplied by another function, use the product rule: (uv)′ = u′v + uv′.

当双曲函数与另一个函数相乘时，使用乘积法则：(uv)′ = u′v + uv′。

Worked example: Differentiate y = x² sinh x.

例题：求 y = x² sinh x 的导数。

Let u = x², v = sinh x. Then u′ = 2x, v′ = cosh x.

令 u = x²，v = sinh x。则 u′ = 2x，v′ = cosh x。

dy/dx = 2x sinh x + x² cosh x

Always keep the original functions intact when applying the product rule to avoid simplification errors.

应用乘积法则时，始终保留原函数，避免化简错误。

6. Quotient Rule with Hyperbolic Functions | 双曲函数的商法则

For quotients involving hyperbolic functions, use the quotient rule: (u/v)′ = (u′v − uv′)/v².

对于涉及双曲函数的商，使用商法则：(u/v)′ = (u′v − uv′)/v²。

Worked example: Differentiate y = tanh x / x.

例题：求 y = tanh x / x 的导数。

Let u = tanh x, v = x. Then u′ = sech² x, v′ = 1.

令 u = tanh x，v = x。则 u′ = sech² x，v′ = 1。

dy/dx = (x sech² x − tanh x) / x²

Simplify only when necessary. Leaving the derivative in factored form is often acceptable in AQA mark schemes.

仅在必要时化简。在AQA评分标准中，导数保留因式形式通常也是可接受的。

7. Derivatives of Inverse Hyperbolic Functions | 反双曲函数的导数

The inverse hyperbolic functions also appear in A-Level Further Mathematics. Their derivatives are particularly useful in integration.

反双曲函数也出现在A-Level进阶数学中。它们的导数在积分中特别有用。

d/dx (arsinh x) = 1/√(x² + 1)

d/dx (arcosh x) = 1/√(x² − 1), x > 1

d/dx (artanh x) = 1/(1 − x²), |x| < 1

These formulas can be remembered by comparing with inverse trigonometric derivatives, noting where signs and domains differ.

这些公式可以与反三角函数的导数对比记忆，注意符号和定义域的差异。

8. Using Identities to Simplify Before Differentiation | 微分前用恒等式化简

Sometimes an expression can be simplified using hyperbolic identities before differentiating, reducing the workload.

有时在微分前使用双曲恒等式化简表达式，可以减少计算量。

For example, differentiate y = cosh² x − sinh² x. By the identity this equals 1, so dy/dx = 0 directly.

例如，求 y = cosh² x − sinh² x 的导数。由恒等式可知它等于 1，因此直接得到 dy/dx = 0。

More usefully, y = sinh 2x can be expanded to 2 sinh x cosh x, but the derivative is easier as 2 cosh 2x. Recognising standard derivatives is faster than product-rule expansions.

更有用的是，y = sinh 2x 可以展开为 2 sinh x cosh x，但导数用 2 cosh 2x 更简便。识别标准导数比展开乘积法则更快。

Key identities to remember: cosh² x − sinh² x = 1, sinh 2x = 2 sinh x cosh x, cosh 2x = cosh² x + sinh² x.

需要记住的关键恒等式：cosh² x − sinh² x = 1、sinh 2x = 2 sinh x cosh x、cosh 2x = cosh² x + sinh² x。

9. Common Exam Mistakes | 常见考试错误

Confusing cosh x derivative with −sinh x. Remember the minus sign only appears for sech, csch and coth.

将 cosh x 的导数误认为 −sinh x。记住负号只出现在 sech、csch 和 coth 中。
Forgetting the chain rule for arguments like 2x or x² . Always multiply by the derivative of the inner function.

忘记对 2x 或 x² 等自变量使用链式法则。务必乘以内层函数的导数。
Using the product rule when the expression is a quotient, or vice versa. Always check the form.

在表达式是商时误用乘积法则，反之亦然。务必先判断形式。
Writing sech² x as 1/cosh² x incorrectly. Since sech x = 1/cosh x, sech² x = 1/cosh² x is correct, but do not write sech² x = 1/sinh² x.

错误地写 sech² x = 1/cosh² x。由于 sech x = 1/cosh x，sech² x = 1/cosh² x 是正确的，但不要写成 sech² x = 1/sinh² x。

10. Worked Exam-Style Problem | 考试风格例题

Question: Differentiate y = e²ˣ tanh(3x).

题目：求 y = e²ˣ tanh(3x) 的导数。

Method: This is a product of e²ˣ and tanh(3x). Apply the product rule.

方法：这是 e²ˣ 与 tanh(3x) 的乘积，应用乘积法则。

Let u = e²ˣ, v = tanh(3x). Then u′ = 2e²ˣ and v′ = 3 sech²(3x).

令 u = e²ˣ，v = tanh(3x)。则 u′ = 2e²ˣ，v′ = 3 sech²(3x)。

dy/dx = 2e²ˣ tanh(3x) + e²ˣ · 3 sech²(3x)

Factor out e²ˣ:

提取公因子 e²ˣ：

dy/dx = e²ˣ [2 tanh(3x) + 3 sech²(3x)]

Check that the derivative of tanh(3x) includes the factor 3 from the chain rule. This is a frequent trap.

注意 tanh(3x) 的导数包含链式法则的因子 3，这是一个常见陷阱。

11. Practice Questions | 练习题目

Attempt these before checking your answers.

先尝试这些题目，再核对答案。

Differentiate y = sinh(5x)

求 y = sinh(5x) 的导数
Differentiate y = cosh x · sin x

求 y = cosh x · sin x 的导数
Differentiate y = ln(cosh x)

求 y = ln(cosh x) 的导数
Differentiate y = tanh⁻¹(2x) (inverse function)

求 y = tanh⁻¹(2x) 的导数（反函数）

Answers:

答案：

5 cosh(5x)

5 cosh(5x)
sinh x sin x + cosh x cos x

sinh x sin x + cosh x cos x
tanh x (since d/dx [ln u] = u′/u = sinh x / cosh x = tanh x)

tanh x（因为 d/dx [ln u] = u′/u = sinh x / cosh x = tanh x）
2/(1 − 4x²)

2/(1 − 4x²)

12. Summary and Revision Tips | 总结与复习建议

The six key derivatives are:

六个关键导数是：

y	dy/dx
sinh x	cosh x
cosh x	sinh x
tanh x	sech² x
coth x	−csch² x
sech x	−sech x tanh x
csch x	−csch x coth x

When differentiating any hyperbolic expression, first identify the outer function, apply its derivative, then multiply by the derivative of the inner function (chain rule). For products and quotients, apply the corresponding rule systematically.

对任意双曲函数表达式求导时，先识别外层函数，应用其导数，再乘以内层函数的导数（链式法则）。对于乘积和商，系统性地应用相应法则。

Practice at least one past-paper question on hyperbolic differentiation each week. Treat hyperbolic functions like their trigonometric counterparts, but pay careful attention to the signs and the absence of periodicity.

每周至少练习一道关于双曲函数微分的真题。将双曲函数视为与三角函数类似，但要特别注意符号以及它们没有周期性。

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Differentiation of Hyperbolic Functions | 双曲函数的微分