📚 Differentiation of Hyperbolic Functions | 双曲函数求导
Hyperbolic functions appear throughout AQA Further Pure and calculus topics. This article explains how to differentiate sinh x, cosh x, tanh x, reciprocal hyperbolic functions and inverse hyperbolic functions, with the chain, product and quotient rules built in.
双曲函数在 AQA Further Pure 和微积分专题中经常出现。本文讲解如何对 sinh x、cosh x、tanh x、倒数双曲函数以及反双曲函数求导，并结合链式法则、乘积法则和商法则。
1. Hyperbolic Functions at a Glance | 双曲函数速览
The hyperbolic functions are defined using the exponential function. The three main definitions are:
双曲函数通过指数函数定义。三个主要定义如下：
sinh x = (eˣ − e⁻ˣ) / 2, cosh x = (eˣ + e⁻ˣ) / 2, tanh x = sinh x / cosh x
Because these functions are built from exponentials, their derivatives follow very clean patterns. You should know these results confidently before attempting exam questions that combine hyperbolic functions with the chain rule.
由于这些函数由指数函数构成，它们的导数规律非常清晰。在开始做结合双曲函数与链式法则的考试题之前，你应当熟练掌握这些结果。
2. Differentiating sinh x from Its Definition | 从定义推导 sinh x 的导数
Start from the exponential definition of sinh x. Differentiate term by term, remembering that the derivative of e⁻ˣ is −e⁻ˣ.
从 sinh x 的指数定义出发，逐项求导，记住 e⁻ˣ 的导数是 −e⁻ˣ。
d/dx (sinh x) = d/dx [(eˣ − e⁻ˣ)/2] = (eˣ + e⁻ˣ)/2 = cosh x
ct parallel to the trigonometric result d/dx (sin x) = cos x, but without any sign change.
因此 sinh x 的导数是 cosh x。这与三角函数结论 d/dx (sin x) = cos x 非常相似，但没有符号变化。
3. Differentiating cosh x and Sign Matters | 对 cosh x 求导及符号问题
Now differentiate the exponential definition of cosh x. The derivative of eˣ is eˣ and the derivative of e⁻ˣ is −e⁻ˣ, so the minus sign becomes a plus sign in the numerator.
现在对 cosh x 的指数定义求导。eˣ 的导数是 eˣ，e⁻ˣ 的导数是 −e⁻ˣ，因此分子中的减号变为加号。
d/dx (cosh x) = d/dx [(eˣ + e⁻ˣ)/2] = (eˣ − e⁻ˣ)/2 = sinh x
cos x) = −sin x. This is one of the most common sign errors in hyperbolic differentiation.
关键结论是 d/dx (cosh x) = sinh x。与 d/dx (cos x) = −sin x 不同，这里没有负号。这是双曲函数求导中最常见的符号错误之一。
4. Quotient Rule for tanh x | 用商法则求 tanh x 的导数
anh x as sinh x / cosh x and apply the quotient rule. If u = sinh x and v = cosh x, then u' = cosh x and v' = sinh x.
将 tanh x 写成 sinh x / cosh x，并应用商法则。设 u = sinh x，v = cosh x，则 u' = cosh x，v' = sinh x。
d/dx (tanh x) = (cosh x cosh x − sinh x sinh x) / cosh²x = (cosh²x − sinh²x) / cosh²x
, this simplifies to 1 / cosh²x, which is sech²x. You may also see it written as 1 − tanh²x.
利用恒等式 cosh²x − sinh²x = 1，可化简为 1 / cosh²x，即 sech²x。它也可以写成 1 − tanh²x。
d/dx (tanh x) = sech²x
Note that sech²x is never negative, which reflects the fact that tanh x is an increasing function on its entire domain.
注意 sech²x 始终非负，这正好说明 tanh x 在其整个定义域上单调递增。
5. Reciprocal Hyperbolic Functions | 倒数双曲函数
/ sinh x and coth x = 1 / tanh x. Their derivatives all carry negative signs in some form.
倒数双曲函数为 sech x = 1 / cosh x、cosech x = 1 / sinh x 和 coth x = 1 / tanh x。它们的导数都带有某种形式的负号。
d/dx (sech x) = −sech x tanh x
d/dx (cosech x) = −cosech x coth x
d/dx (coth x) = −cosech²x
or example, sech x = (cosh x)⁻¹, so its derivative is −(cosh x)⁻² sinh x = −sech x tanh x.
这些结果可由倒数法则或商法则推导。例如 sech x = (cosh x)⁻¹，所以其导数为 −(cosh x)⁻² sinh x = −sech x tanh x。
6. Chain Rule in Action | 链式法则的实际应用
In exam questions you will usually differentiate composite functions such as sinh(ax + b), cosh(kx) or tanh(3x). Multiply the outer derivative by the derivative of the inner linear function.
考试中通常需要求复合函数如 sinh(ax + b)、cosh(kx) 或 tanh(3x) 的导数。将外层导数乘以内层线性函数的导数即可。
d/dx [sinh(kx)] = k cosh(kx)
d/dx [cosh(ax + b)] = a sinh(ax + b)
d/dx [tanh(3x)] = 3 sech²(3x)
If the inner function is not linear, for example sinh(x² + 1), the chain rule gives 2x cosh(x² + 1). Always write the inner derivative clearly to avoid missing a factor.
如果内层函数不是线性的，例如 sinh(x² + 1)，链式法则给出 2x cosh(x² + 1)。始终把内层导数写清楚，以免漏掉因子。
7. Product and Quotient Rules | 乘积法则与商法则
Hyperbolic functions often appear in products or quotients with algebraic or exponential functions. The product rule and quotient rule work exactly as usual.
双曲函数经常与代数函数或指数函数以乘积或商的形式出现。乘积法则和商法则照常适用。
Example 1: if y = x sinh x, then u = x and v = sinh x, so u' = 1 and v' = cosh x. Therefore dy/dx = sinh x + x cosh x.
例 1：若 y = x sinh x，则 u = x，v = sinh x，因此 u' = 1，v' = cosh x。所以 dy/dx = sinh x + x cosh x。
Example 2: if y = eˣ cosh x, then dy/dx = eˣ cosh x + eˣ sinh x = eˣ(cosh x + sinh x) = eˣ·eˣ = e²ˣ, using cosh x + sinh x = eˣ.
例 2：若 y = eˣ cosh x，则 dy/dx = eˣ cosh x + eˣ sinh x = eˣ(cosh x + sinh x) = eˣ·eˣ = e²ˣ，这里利用了 cosh x + sinh x = eˣ。
quotient rule gives (x cosh x − sinh x) / x². State which rule you are using in structured answers.
对于商，例如 y = sinh x / x，商法则给出 (x cosh x − sinh x) / x²。在结构化解题中，请说明你使用的是哪条法则。
8. Inverse Hyperbolic Functions | 反双曲函数
The inverse hyperbolic functions are arsinh x, arcosh x and artanh x. They are also written as sinh⁻¹x, cosh⁻¹x and tanh⁻¹x in AQA notation, but this is not a reciprocal power.
反双曲函数包括 arsinh x、arcosh x 和 artanh x。在 AQA 记号中也写作 sinh⁻¹x、cosh⁻¹x 和 tanh⁻¹x，但这并不表示倒数幂。
Their derivatives are:
它们的导数如下：
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
tiate implicitly: cosh y dy/dx = 1. Since cosh²y = 1 + sinh²y = 1 + x², we get dy/dx = 1 / √x² + 1.
推导第一个结果：令 y = arsinh x，则 sinh y = x。隐函数求导得 cosh y dy/dx = 1。由于 cosh²y = 1 + sinh²y = 1 + x²，所以 dy/dx = 1 / √x² + 1。
The chain rule version is also common: d/dx [artanh(3x)] = 3 / (1 − 9x²), provided |3x| < 1.
链式法则版本也很常见：d/dx [artanh(3x)] = 3 / (1 − 9x²)，其中要求 |3x| < 1。
9. Second Derivatives and Differential Equations | 二阶导数与微分方程
ves return to the original functions. This makes hyperbolic functions natural solutions of certain differential equations.
因为 d/dx (sinh x) = cosh x 且 d/dx (cosh x) = sinh x，所以二阶导数会回到原函数。这使得双曲函数成为某些微分方程的自然解。
d²/dx² (sinh x) = sinh x, d²/dx² (cosh x) = cosh x
For example, y = A cosh x + B sinh x satisfies the differential equation y'' − y = 0. This is analogous to the trigonometric solution for y'' + y = 0.
例如，y = A cosh x + B sinh x 满足微分方程 y'' − y = 0。这类似于三角函数对 y'' + y = 0 的解。
If you are asked to verify a solution, differentiate twice and substitute back into the equation. Show each derivative separately to gain method marks.
如果题目要求验证某个解，请连续求两次导并代回方程。把每一步导数分开写，以便获得方法分。
10. Worked Exam-Style Examples | 考试型例题精讲
sinh²(3x). Let u = sinh(3x), so y = u². Then dy/dx = 2u × 3cosh(3x) = 6 sinh(3x) cosh(3x).
典型例题 1：对 y = sinh²(3x) 求导。设 u = sinh(3x)，则 y = u²。因此 dy/dx = 2u × 3cosh(3x) = 6 sinh(3x) cosh(3x)。
le 2: Differentiate y = x tanh(2x). Use the product rule: dy/dx = tanh(2x) + 2x sech²(2x).
典型例题 2：对 y = x tanh(2x) 求导。使用乘积法则：dy/dx = tanh(2x) + 2x sech²(2x)。
Worked example 3: Differentiate y = arcosh(4x). The chain rule gives dy/dx = 4 / √16x² − 1, valid for x > 1/4.
典型例题 3：对 y = arcosh(4x) 求导。链式法则给出 dy/dx = 4 / √16x² − 1，其有效范围为 x > 1/4。
11. Common Mistakes to Avoid | 常见错误与避坑指南
Many marks are lost through small sign slips. The derivative of cosh x is positive sinh x, not −sinh x. The derivative of tanh x is sech²x, not −sech²x.
许多失分来自细小的符号错误。cosh x 的导数是正的 sinh x，而不是 −sinh x。tanh x 的导数是 sech²x，而不是 −sech²x。
coth x all have a negative sign in their derivatives. Remember: d/dx (coth x) = −cosech²x.
对于倒数双曲函数，sech x、cosech x 和 coth x 的导数都带有负号。请记住：d/dx (coth x) = −cosech²x。
When differentiating inverse hyperbolic functions, do not confuse a reciprocal power with the inverse function. Also check the domain: arcosh x requires x > 1, and artanh x requires |x| < 1.
对反双曲函数求导时，不要把倒数幂与反函数混淆。同时要检查定义域：arcosh x 需要 x > 1，artanh x 需要 |x| < 1。
12. Summary and Key Results | 总结与核心公式清单
The table below collects the essential derivatives you should memorise for AQA examination questions.
下表汇总了你在 AQA 考试中需要记住的核心导数公式。
| Function | Derivative | Notes |
|---|---|---|
| sinh x | cosh x | No sign change |
| cosh x | sinh x | Positive, not negative |
| tanh x | sech²x | Also 1 − tanh²x |
| sech x | −sech x tanh x | Negative sign |
| cosech x | −cosech x coth x | Negative sign |
| coth x | −cosech²x | Negative sign |
Practise applying these with the chain rule until the algebra feels automatic. Then tackle mixed questions involving products, quotients, implicit differentiation and inverse hyperbolic functions.
将这些公式与链式法则结合反复练习，直到代数过程熟练自如。然后再挑战涉及乘积、商、隐函数求导和反双曲函数的综合题。
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