Derivatives of Alternative Functions | 其他函数的导数

📚 Derivatives of Alternative Functions | 其他函数的导数

In IB Mathematics, once you master differentiating power functions, you encounter a rich family of ‘alternative’ functions: exponentials, logarithms, trigonometric functions, and their inverses. These derivatives underpin models in growth, oscillation, and optimisation. This article guides you through every essential rule, from eˣ to arctan x, with clear examples, common pitfalls, and practical applications.

在 IB 数学中，掌握了幂函数求导后，你将遇到一系列“其他”函数：指数、对数、三角函数及反三角函数。这些函数的导数是增长、振荡和优化模型的基础。本文将带你逐一掌握从 eˣ 到 arctan x 的核心求导法则，配合清晰示例、常见错误提示与实际应用。

1. Quick Recap: Differentiation Fundamentals | 求导基础回顾

Before diving into alternative functions, recall the basic rules. The power rule states that d/dx (xⁿ) = n xⁿ⁻¹ for any real n. The sum rule, constant multiple rule, product rule, quotient rule, and chain rule form the toolkit we will extend. The chain rule is especially vital: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).

在进入其他函数之前，先回顾基本法则。幂法则：d/dx (xⁿ) = n xⁿ⁻¹（n 为任意实数）。和法则、常数倍法则、乘法法则、除法法则和链式法则构成了我们将要扩展的工具箱。链式法则尤为重要：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。

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2. The Exponential Function eˣ | 指数函数 eˣ

The exponential function f(x) = eˣ is remarkably its own derivative: d/dx (eˣ) = eˣ. More generally, if the exponent is a function u(x), the chain rule gives d/dx (eᵘ) = eᵘ·u'(x). For example, differentiate y = e²ˣ: let u = 2x → u’ = 2, so dy/dx = e²ˣ·2 = 2 e²ˣ.

指数函数 f(x) = eˣ 的独特之处在于它等于自己的导数：d/dx (eˣ) = eˣ。更一般地，若指数是函数 u(x)，链式法则给出 d/dx (eᵘ) = eᵘ·u'(x)。例如求导 y = e²ˣ：令 u = 2x → u’ = 2，则 dy/dx = e²ˣ·2 = 2 e²ˣ。

The simplicity of eˣ arises from the natural base e ≈ 2.718, which makes it the foundation of continuous growth and decay models.

eˣ 的简洁性源于自然底数 e ≈ 2.718，这使其成为连续增长与衰减模型的基础。

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3. General Exponential Function aˣ | 一般指数函数 aˣ

For a base a > 0, a ≠ 1, the derivative is d/dx (aˣ) = aˣ ln a. This formula emerges from rewriting aˣ = e^(x ln a) and applying the chain rule. Example: d/dx (2ˣ) = 2ˣ ln 2. With a function in the exponent, d/dx (aᵘ) = aᵘ ln a · u'(x).

对于底数 a > 0 且 a ≠ 1，导数为 d/dx (aˣ) = aˣ ln a。该公式源于将 aˣ 改写为 e^(x ln a) 并应用链式法则。示例：d/dx (2ˣ) = 2ˣ ln 2。若指数是函数，则 d/dx (aᵘ) = aᵘ ln a · u'(x)。

Memorise this pattern: multiply by the natural log of the base, then by the derivative of the exponent.

记忆模式：乘以底数的自然对数，再乘以指数的导数。

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4. Natural Logarithmic Function ln x | 自然对数函数 ln x

The derivative of f(x) = ln x (for x>0) is d/dx (ln x) = 1/x. For a more complex argument u(x) > 0, the chain rule yields d/dx (ln u) = (1/u)·u'(x) = u’/u. Example: y = ln(3x²+1) gives dy/dx = (6x)/(3x²+1).

自然对数 f(x) = ln x（x>0）的导数为 d/dx (ln x) = 1/x。若真数为函数 u(x)>0，链式法则给出 d/dx (ln u) = (1/u)·u'(x) = u’/u。例如：y = ln(3x²+1) → dy/dx = (6x)/(3x²+1)。

When the argument can be negative, the derivative of ln|x| is also 1/x, a useful extension in integration.

当真数可能为负时，ln|x| 的导数也是 1/x，这一推广在积分中非常有用。

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5. Logarithm with General Base logₐ x | 一般底数的对数函数 logₐ x

Using the change-of-base formula logₐ x = (ln x)/(ln a), we obtain d/dx (logₐ x) = 1/(x ln a). For a composite argument, d/dx (logₐ u) = u’/(u ln a). Example: d/dx (log₂(5x)) = (5)/(5x ln 2) = 1/(x ln 2).

利用换底公式 logₐ x = (ln x)/(ln a)，可得 d/dx (logₐ x) = 1/(x ln a)。对于复合变量，d/dx (logₐ u) = u’/(u ln a)。示例：d/dx (log₂(5x)) = (5)/(5x ln 2) = 1/(x ln 2)。

Notice the pattern: the derivative of any log is essentially 1/(argument) divided by ln(base), times the derivative of the argument.

请注意模式：任何对数函数的导数本质上是 1/(真数) 除以 ln(底数)，再乘以真数的导数。

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6. Derivatives of Sine and Cosine | 正弦与余弦函数的导数

The basic trigonometric derivatives are d/dx (sin x) = cos x and d/dx (cos x) = -sin x. These are fundamental and must be memorised. With the chain rule: d/dx (sin u) = cos u · u’ and d/dx (cos u) = -sin u · u’. Example: y = sin(2x) gives dy/dx = cos(2x)·2 = 2 cos(2x).

基本三角函数的导数为 d/dx (sin x) = cos x 和 d/dx (cos x) = -sin x。这些是基础，必须熟记。结合链式法则：d/dx (sin u) = cos u · u’，d/dx (cos u) = -sin u · u’。示例：y = sin(2x) → dy/dx = cos(2x)·2 = 2 cos(2x)。

The sign change for cosine derivative is a frequent source of errors — always check it.

余弦导数中的符号变化是常见错误源——务必检查。

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7. Derivatives of Other Trigonometric Functions | 其他三角函数的导数

From sin and cos we derive the rest: d/dx (tan x) = sec² x; d/dx (cot x) = -csc² x; d/dx (sec x) = sec x tan x; d/dx (csc x) = -csc x cot x. These can all be obtained via the quotient rule. For instance, tan x = sin x / cos x → derivative = (cos² x + sin² x)/cos² x = 1/cos² x = sec² x.

由 sin 和 cos 可推导其余：d/dx (tan x) = sec² x；d/dx (cot x) = -csc² x；d/dx (sec x) = sec x tan x；d/dx (csc x) = -csc x cot x。这些均可通过商法则得出。例如 tan x = sin x / cos x → 导数 = (cos² x + sin² x)/cos² x = 1/cos² x = sec² x。

In chain-rule form, each formula includes u’: d/dx (tan u) = sec² u · u’, etc. Practice with y = tan(3x) → dy/dx = 3 sec²(3x).

在链式法则形式中，每个公式都包含 u’：d/dx (tan u) = sec² u · u’，等等。练习：y = tan(3x) → dy/dx = 3 sec²(3x)。

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

IB syllabi often include derivatives of arcsin, arccos, and arctan. The key formulas are: d/dx (arcsin x) = 1/√(1 – x²), d/dx (arccos x) = -1/√(1 – x²), d/dx (arctan x) = 1/(1 + x²). All are valid for appropriate domains. With chain rule: d/dx (arcsin u) = u’/√(1 – u²).

IB 课程通常涵盖 arcsin、arccos 和 arctan 的导数。关键公式为：d/dx (arcsin x) = 1/√(1 – x²)，d/dx (arccos x) = -1/√(1 – x²)，d/dx (arctan x) = 1/(1 + x²)。所有公式均在相应定义域内成立。链式法则形式：d/dx (arcsin u) = u’/√(1 – u²)。

These derivatives are often proved using implicit differentiation. For example, let y = arcsin x ⇒ sin y = x ⇒ cos y · dy/dx = 1 ⇒ dy/dx = 1/√(1 – x²) because cos y = √(1 – sin² y) = √(1 – x²).

这些导数常用隐函数求导证明。例如，设 y = arcsin x ⇒ sin y = x ⇒ cos y · dy/dx = 1 ⇒ dy/dx = 1/√(1 – x²)，因为 cos y = √(1 – sin² y) = √(1 – x²)。

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9. Chain Rule with Mixed Functions | 链式法则与混合函数

Real problems combine algebraic and alternative functions. Example: Differentiate y = e^(sin x) · ln(x²+1). We need product rule and chain rule. Let u = e^(sin x), v = ln(x²+1). u’ = e^(sin x) · cos x; v’ = (2x)/(x²+1). Then dy/dx = u’ v + u v’. Such layered differentiation is common in exams.

实际问题常将代数函数与其他函数混合。示例：对 y = e^(sin x) · ln(x²+1) 求导。需要使用乘法法则和链式法则。令 u = e^(sin x)，v = ln(x²+1)。u’ = e^(sin x) · cos x；v’ = (2x)/(x²+1)。则 dy/dx = u’ v + u v’。这类分层求导在考试中十分常见。

Always identify the outermost operation first, then work inward, keeping track of each derivative factor.

始终先识别最外层运算，然后向内推进，并追踪每个导数因子。

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10. Logarithmic Differentiation | 对数求导法

When a function has a variable in both base and exponent, like y = x^(sin x), take natural logs: ln y = sin x · ln x. Differentiate implicitly: (1/y) dy/dx = cos x · ln x + sin x · (1/x). Then dy/dx = y [cos x ln x + (sin x)/x] = x^(sin x) [cos x ln x + sin x / x].

当函数底数和指数均含变量时，如 y = x^(sin x)，可先取自然对数：ln y = sin x · ln x。隐函数求导得：(1/y) dy/dx = cos x · ln x + sin x · (1/x)。于是 dy/dx = y [cos x ln x + (sin x)/x] = x^(sin x) [cos x ln x + sin x / x]。

Logarithmic differentiation also simplifies products or quotients of several functions, turning them into sums before differentiating.

对数求导法还能简化多个函数乘积或商的求导，在求导前将其转化为和。

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11. Higher-Order Derivatives | 高阶导数

Alternative functions produce elegant higher-order patterns. For y = eᵏˣ, the nth derivative is kⁿ eᵏˣ. For y = sin x, derivatives cycle every four steps: sin x → cos x → -sin x → -cos x → sin x. For y = ln x, y” = -1/x², y”’ = 2/x³, etc. Understanding these cycles is useful for Taylor series.

其他函数会生成优雅的高阶导数模式。对 y = eᵏˣ，n 阶导数为 kⁿ eᵏˣ。对 y = sin x，导数每四步循环一次：sin x → cos x → -sin x → -cos x → sin x。对 y = ln x，y” = -1/x²，y”’ = 2/x³，等等。理解这些周期对泰勒级数很有用。

When finding the second derivative in implicit or parametric contexts, treat dy/dx as a new function and differentiate again, carefully applying the chain rule.

在隐函数或参数方程中求二阶导数时，将 dy/dx 视为新函数再次求导，并小心应用链式法则。

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12. Summary and Formula Table | 总结与公式表

Below is a concise table of essential derivatives of alternative functions that every IB student should memorise.

下表总结了每位 IB 学生都应牢记的其他函数导数公式。

Function f(x)	Derivative f'(x)	Chain Rule Version f'(u)·u’
eˣ	eˣ	eᵘ · u’
aˣ	aˣ ln a	aᵘ ln a · u’
ln x	1/x	u’/u
logₐ x	1/(x ln a)	u’/(u ln a)
sin x	cos x	cos u · u’
cos x	-sin x	-sin u · u’
tan x	sec² x	sec² u · u’
arcsin x	1/√(1 – x²)	u’/√(1 – u²)
arccos x	-1/√(1 – x²)	-u’/√(1 – u²)
arctan x	1/(1 + x²)	u’/(1 + u²)

Master these derivatives, combine them confidently with the product, quotient, and chain rules, and you will tackle a huge range of IB calculus problems effectively.

掌握这些导数公式，自信地与乘法、除法和链式法则相结合，你就能高效解决 IB 微积分中的大量问题。

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