Derivatives: A Table of Standard Derivatives | 导数：标准导数表

📚 Derivatives: A Table of Standard Derivatives | 导数：标准导数表

Differentiation is one of the central pillars of calculus, and having a ready reference of standard derivatives is essential for any IB Mathematics student. This article presents a comprehensive table of derivatives together with explanations of the underlying rules, from the limit definition to the derivatives of exponential, logarithmic, trigonometric and inverse trigonometric functions. We also cover the chain rule, product rule and quotient rule, as these are needed to differentiate composite functions. Worked examples illustrate how to use the table in practice, making this a complete revision resource.

微分是微积分的核心支柱之一，拥有一份标准导数速查表对每一位 IB 数学学生都至关重要。本文既给出系统的导数表，也解释背后的法则，包括极限定义、指数函数、对数函数、三角函数和反三角函数的导数。我们还会讲解链式法则、积法则与商法则，因为它们是求复合函数导数的基础。文末配有例题，展示如何将导数表用于实际计算，为你提供一份完整的复习资料。

---

1. What is a Derivative? | 什么是导数？

x) at a point x is defined as the limit of the difference quotient as h approaches zero: f'(x) = lim_h→0 [f(x+h) – f(x)] / h. It measures the instantaneous rate of change or the slope of the tangent line to the graph of f.

函数 f(x) 在某一点 x 处的导数定义为差商的极限，当 h 趋近于 0 时：f'(x) = lim_h→0 [f(x+h) – f(x)] / h。它衡量的是瞬时变化率，也就是函数图像在该点切线的斜率。

If this limit exists, we say f is differentiable at x. The process of finding a derivative is called differentiation. The notations f'(x), dy/dx, and d/dx[f(x)] are all commonly used in IB examinations.

如果这个极限存在，我们就说 f 在 x 处可导。求导数的过程称为微分。在 IB 考试中，f'(x)、dy/dx 和 d/dx[f(x)] 都是常见的记号。

---

2. The Derivative as a Limit | 导数作为极限

rivatives of basic functions. For a constant function f(x) = c, the difference quotient [c – is is the first entry in our table of derivatives: the derivative of any constant is zero.

利用极限定义，我们可以推导出基本函数的导数。对于常数函数 f(x) = c，差商 [c – c]/h = 0，所以 d/dx (c) = 0。这是导数表中的第一条：任意常数的导数为零。

For f(x) = x, the difference quotient is [ (x+h) – x ]/h = h/h = 1, giving f' nomial expansion of (x+h)ⁿ, but the limit definition underpins every formula in the table.

对于 f(x) = x，差商是 [ (x+h) – x ]/h = h/h = 1，因此 f'(x) = 1。更一般地，幂函数求导法则可以从 (x+h)ⁿ 的二项展开式中得到，但极限定义是表中所有公式的根基。

---

3. Basic Differentiation Rules | 基本微分法则

If f and g are differentiable functions and k is a constant, then: d/dx [f(x) ± g(x)] = f'(x) ± g'(x); and d/dx [k f(x)] = k f'(x). These linearity properties make differentiation easier.

在充实导数表之前，我们先给出组合函数求导的法则。如果 f 和 g 是可微函数，k 是常数，那么：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)；并且 d/dx [k f(x)] = k f'(x)。这些线性性质让求导变得更加简单。

For example, d/dx (3x² + 5x – se rules, we would have to return to the limit definition each time, which is impractical.

例如，d/dx (3x² + 5x – 7) = 3·2x + 5·1 + 0 = 6x + 5。如果没有这些法则，每次都要回到极限定义，那是不切实际的。

---

4. Power Rule | 幂函数求导法则

The power rule states that for any real number n, the derivative of xⁿ is n xⁿ⁻¹. This can be written as:

幂函数求导法则指出，对任意实数 n，xⁿ 的导数是 n xⁿ⁻¹。可以写成：

d/dx (xⁿ) = n xⁿ⁻¹

functions, including polynomials (n positive integer), roots (n = 1/2 gives derivative of √x), and reciprocals (n = –1 gives derivative of 1/x). When n = 1/2, f(x) = x^1/2 = √x, and f'(x) = (1/2) x^–1/2 = 1/(2√x).

这个公式涵盖了大量函数，包括多项式（n 为正整数）、根式（n=1/2 得到 √x 的导数）以及倒数（n=–1 得到 1/x 的导数）。当 n = 1/2 时，f(x) = x^1/2 = √x，f'(x) = (1/2) x^–1/2 = 1/(2√x)。

For n = –1, f(x) = x⁻¹ = 1/x, so f'(x) = –1 x⁻² = – e power rule is arguably the most frequently used entry in the standard derivatives table.

当 n = –1 时，f(x) = x⁻¹ = 1/x，f'(x) = –1 x⁻² = –1/x²。幂函数求导法则可以说是标准导数表中最常用的一条。

---

5. Derivatives of Exponential Functions | 指数函数的导数

The natural exponential function f(x) = eˣ has the remarkable property that its derivative is itself: d/dx (eˣ) = eˣ. More generally, for a > 0, d/dx (aˣ) = aˣ ln a. This follows from writing aˣ = e^{x ln a} and applying the chain rule.

自然指数函数 f(x) = eˣ 有一个非常优美的性质：它的导数等于它自身，即 d/dx (eˣ) = eˣ。更一般地，对 a > 0，d/dx (aˣ) = aˣ ln a。这是因为可以把 aˣ 写成 e^{x ln a} 再使用链式法则。

Thus, the derivative of 2ˣ is 2ˣ n multiplied by a constant factor. For the natural exponential, ln e = 1, simplifying to eˣ.

因此，2ˣ 的导数是 2ˣ ln 2。注意 ln a 是常数，所以结果相当于原函数乘以一个常数因子。对于自然指数，ln e = 1，化简后就是 eˣ。

---

6. Derivatives of Logarithmic Functions | 对数函数的导数

s result can be obtained by implicit differentiation of y = ln x, which is equivalent to eʸ = x. More generally, for a > 0 and a ≠ 1, d/dx (logₐ x) = 1/(x ln a).

自然对数的导数是 d/dx (ln x) = 1/x，其中 x > 0。这个结果可以通过对 y = ln x 进行隐函数求导得到，它等价于 eʸ = x。更一般地，对 a > 0 且 a ≠ 1，d/dx (logₐ x) = 1/(x ln a)。

The restriction x > 0 is important because the logarithm is only defined for positive arguments. When differentiating ln|u|, where u is a function of x, the derivative is u'/u by the chain rule, which works for u > 0 or u < 0.

x > 0 的限制很重要，因为对数只对正的自变量有定义。当对 ln|u| 求导时（u 是 x 的函数），由链式法则得到导数为 u'/u，这一结果对 u > 0 或 u < 0 都成立。

---

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

The derivatives of the six basic trigonometric functions are essential entries in the table. They are summarised as follows, with x in radians:

六个基本三角函数的导数是导数表中的核心内容。它们总结如下，x 的单位为弧度：

d/dx (sin x) = cos x
d/dx (cos x) = –sin x
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

long with trigonometric identities. For example, d/dx (sin x) = cos x can be shown via lim_h→0 (sin(x+h) – sin x)/h and the sine addition formula. The derivative of cos x yields the negative sign.

这些公式是通过极限定义和三角恒等式推导出来的。例如，利用 lim_h→0 (sin(x+h) – sin x)/h 以及正弦的和角公式可以证明 d/dx (sin x) = cos x。对 cos x 求导则会出现负号。

---

广告 / Sponsored

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

Inverse trigonometric functions also appear in the IB syllabus, particularly at Higher Level. Their derivatives are obtained using implicit differentiation. The most commonly required ones are:

反三角函数也出现在 IB 数学大纲中，尤其是在高水平课程里。它们的导数可以通过隐函数求导得到。最常要求的几个是：

d/dx (arcsin x) = 1/√1 – x²
d/dx (arccos x) = –1/√1 – x²
d/dx (arctan x) = 1/(1 + x²)

ferentiating implicitly with respect to x gives 1 = cos y · dy/dx, so dy/dx = 1/cos y = 1/√1 – sin² y = 1/√1 – x². The domains are |x| < 1 for arcsin and arccos, while arctan is defined for all real x.

例如，令 y = arcsin x，则 x = sin y。对 x 隐式求导得 1 = cos y · dy/dx，因此 dy/dx = 1/cos y = 1/√1 – sin² y = 1/√1 – x²。arcsin 和 arccos 的定义域要求 |x| < 1，而 arctan 对所有实数 x 都有定义。

---

9. The Chain Rule | 链式法则

he chain rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) · g' du/dx). This rule extends the standard derivatives table to compositions like sin(kx) or e^x².

链式法则用于求复合函数的导数。如果 y = f(g(x))，那么 dy/dx = f'(g(x)) · g'(x)。在莱布尼茨记号下，如果 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) · (du/dx)。这一法则将标准导数表推广到诸如 sin(kx) 或 e^x² 这样的复合函数。

y/dx = 3 cos(3x). Without the chain rule, our table would only handle bare functions of x.

例如，对 y = sin(3x) 求导，我们可以令 u = 3x，则 y = sin u。于是 dy/du = cos u，du/dx = 3，最终 dy/dx = 3 cos(3x)。如果没有链式法则，导数表就只限于直接对 x 的函数求导。

---

10. Product and Quotient Rules | 积法则与商法则

ifferentiating the product of two functions, we use the product rule: d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x). For the quotient, the rule is: d/dx [u(x)/v(x)] = [u'(x) v(x) – u(x) v'(x)] / [v(x)]², provided v(x) ≠ 0.

对两个函数的乘积求导时，我们使用积法则：d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)。对于商，法则是：d/dx [u(x)/v(x)] = [u'(x) v(x) – u(x) v'(x)] / [v(x)]²，前提是 v(x) ≠ 0。

These rules can be memorised in words: the derivative of a product is the derivative of the first times the second, plus the first times the derivative of the second. For the quotient, it is the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared.

这些法则可以用语言来记忆：乘积的导数等于“前导后不导加后导前不导”。商的导数则是“上导下不导减下导上不导，除以分母的平方”。

---

11. Summary Table of Standard Derivatives | 标准导数汇总表

The table below lists the most commonly used derivatives in IB Mathematics. All entries assume x is the independent variable, and functions are differentiable on their domains.

下表列出了 IB 数学中最常用的导数。所有函数均以 x 为自变量，并假定在定义域内可导。

f(x)	f'(x)
c (constant)	0
xⁿ	n xⁿ⁻¹
eˣ	eˣ
aˣ (a > 0)	aˣ ln a
ln x	1/x
logₐ x	1/(x ln a)
sin x	cos x
cos x	–sin x
tan x	sec² x
cot x	–csc² x
sec x	sec x tan x
csc x	–csc x cot x
arcsin x	1/√1 – x²
arccos x	–1/√1 – x²
arctan x	1/(1 + x²)

Combined with the chain rule, product rule and quotient rule, this table allows you to differentiate almost any elementary function encountered in the IB course.

结合链式法则、积法则和商法则，这张表格可以帮助你求出 IB 课程中几乎所有初等函数的导数。

---

12. Worked Examples | 例题精解

Example 1: Differentiate f(x) = 5x⁴ – 3eˣ + 2 ln x. Using the table and the sum/constant multiple rules: f'(x) = 5·4x³ – 3·eˣ + 2·(1/x) = 20x³ – 3eˣ + 2/x.

例1：对 f(x) = 5x⁴ – 3eˣ + 2 ln x 求导。利用导数表以及和/常数倍法则：f'(x) = 5·4x³ – 3·eˣ + 2·(1/x) = 20x³ – 3eˣ + 2/x。

Example 2: Differentiate g(x) = (x² + 1) sin x. This is a product: u = x² + 1, v = sin x. Then u' = 2x, v' = cos x. By the product rule: g'(x) = 2x sin x + (x² + 1) cos x.

例2：对 g(x) = (x² + 1) sin x 求导。这是一个乘积：u = x² + 1, v = sin x。则 u' = 2x, v' = cos x。由积法则：g'(x) = 2x sin x + (x² + 1) cos x。

Example 3: Differentiate h(x) = e^2x / (x + 3). This requires the quotient rule and chain rule. Let u = e^2x (chain rule gives u' = 2e^2x) and v = x + 3 (v' = 1). Then h'(x) = [2e^2x(x+3) – e^2x·1] / (x+3)² = e^2x(2x + 6 – 1) / (x+3)² = e^2x(2x + 5) / (x+3)².

例3：对 h(x) = e^2x / (x + 3) 求导。这需要商法则和链式法则。令 u = e^2x（链式法则得 u' = 2e^2x），v = x + 3（v' = 1）。则 h'(x) = [2e^2x(x+3) – e^2x·1] / (x+3)² = e^2x(2x + 6 – 1) / (x+3)² = e^2x(2x + 5) / (x+3)²。

Example 4: /du = 1/u, du/dx = cos x. Hence dy/dx = cos x / sin x = cot x. This matches the pattern ln|u| → u'/u, confirming the derivative of ln(sin x) is cot x for sin x > 0.

例4：求 y = ln(sin x) 的导数。使用链式法则：令 u = sin x，则 y = ln u。于是 dy/du = 1/u，du/dx = cos x。因此 dy/dx = cos x / sin x = cot x。这符合 ln|u| → u'/u 的模式，验证了在 sin x > 0 时 ln(sin x) 的导数是 cot x。

---

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）