8. A Basic Table of Derivatives | 8. 基本导数表

📚 8. A Basic Table of Derivatives | 8. 基本导数表

In IB Mathematics, differentiation is a central topic, and a solid command of the basic table of derivatives is essential for solving problems efficiently. This article gathers the derivatives of the most common elementary functions, explaining each entry and showing how to apply it. Whether you are studying Analysis & Approaches SL or HL, having this table at your fingertips will boost both your speed and your understanding of calculus.

在 IB 数学中，微分是一个核心主题，熟练掌握基本导数表对于高效解题至关 重要。本文汇集了最常见基本函数的导数，逐一解释并展示其应用。无论你在学习分析与方法的 SL 还是 HL，将这个表格熟记于心都会提高你的解题速度和微积分理解。

1. What Is a Derivative? | 什么是导数？

nstantaneous rate of change of f with respect to x. Formally, it is defined as the limit f'(x) = lim_h→0 [f(x + h) − f(x)] / h, provided the limit exists. This definition underpins all the formulas in the basic derivative table.

函数 f 在点 x 处的导数表示 f 关于 x 的瞬时变化率。正式定义为极限 f'(x) = lim_h→0 [f(x + h) − f(x)] / h，前提是该极限存在。这个定义是我们基本导数表中所有公式的基础。

Geometrically, the derivative is the slope of the tangent line to the graph of f at (x, f(x)). A positive derivative means the function is increasing, a negative one means it is decreasing, and a zero derivative often indicates a horizontal tangent or a stationary point.

在几何上，导数就是 f 的图像在点 (x, f(x)) 处切线的斜率。导数为正表示函数递增，为负表示递减，为零则通常表示有一条水平切线或驻点。

Common notations for the derivative include f'(x), dy/dx, and d/dx [f(x)]. In IB exams, you are expected to move flexibly among these notations.

导数的常用记号包括 f'(x)、dy/dx 和 d/dx [f(x)]。在 IB 考试中，你应该能灵活切换这些记号。

2. Derivative of a Constant Function | 常数函数的导数

If f(x) = c, where c is any real constant, then f' limit definition: f(x + h) − f(x) = c − c = 0, so the difference quotient is always zero.

若 f(x) = c（c 为任意实常数），则 f'(x) = 0。这可以直接由极限定义得出：f(x + h) − f(x) = c − c = 0，因此差商恒为零。

The graph of a constant function is a horizontal line, and a horizontal line has zero slope everywhere. So the derivative being zero matches the geometric interpretation perfectly.

常数函数的图像是一条水平直线，而水平直线上每一点的斜率都是零。因此导数为零与几何解释完全一致。

d/dx (c) = 0

Example: If f(x) = −7, then f'(x) = 0; if y = 2024, then dy/dx = 0.

例子：若 f(x) = −7，则 f'(x) = 0；若 y = 2024，则 dy/dx = 0。

3. The Power Rule for xⁿ | xⁿ 的幂法则

For any real constant n, the derivative of xⁿ is n·xⁿ⁻¹. This rule works for positive integers, negative integers, and any rational or irrational exponent as long as the function is appropriately defined.

对于任意实常数 n，xⁿ 的导数为 n·xⁿ⁻¹。只要函数有适当的定义，这个法则适用于正整数、负整数，以及任意有理数或无理数指数。

d/dx (xⁿ) = n xⁿ⁻¹

A classic example is x³: here n = 3, so d/dx (x³) = 3x². For x⁻², we get −2x⁻³, which is −2/x³. The power rule also handles roots: √x = x^1/2, so d/dx (√x) = (1/2)x^−1/2 = 1/(2√x).

一个经典例子是 x³：这里 n = 3，所以 d/dx (x³) = 3x²。对于 x⁻²，我们得到 −2x⁻³，即 −2/x³。幂法则同样能处理根式：√x = x^1/2，因此 d/dx (√x) = (1/2)x^−1/2 = 1/(2√x)。

When differentiating a constant multiple times a power, the constant simply stays: d/dx (5x⁴) = 5·4x³ = 20x³. This combination of constant multiple rule and power rule is used very frequently.

当对常数乘以幂函数求导时，常数直接保留：d/dx (5x⁴) = 5·4x³ = 20x³。这种常数倍法则与幂法则的结合极为常用。

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

The natural exponential function e^x has the remarkable property that its derivative is itself: d/dx (e^x) = e^x. This is why e ≈ 2.718 is the natural base for calculus.

自然指数函数 e^x 具有一个显著的性质：它的导数就是它本身：d/dx (e^x) = e^x。这正是 e ≈ 2.718 被选为微积分自然底数的原因。

d/dx (e^x) = e^x

For a general exponential function with base a > 0 and a ≠ 1, we use the formula d/dx (a^x) = a^x ln a. This can be derived by writing a^x = e^{x ln a} and applying the chain rule.

对于底数为 a（a > 0 且 a ≠ 1）的一般指数函数，我们使用公式 d/dx (a^x) = a^x ln a。这可以通过写成 a^x = e^{x ln a} 并应用链式法则来推导。

d/dx (a^x) = a^x ln a

Example: d/dx (2^x) = 2^x ln 2; d/dx (10^x) = 10^x ln 10. When a = e, ln e = 1, so the formula reduces to e^x·1 = e^x, consistent with the special case.

例子：d/dx (2^x) = 2^x ln 2；d/dx (10^x) = 10^x ln 10。当 a = e 时，ln e = 1，公式化为 e^x·1 = e^x，与特殊情况一致。

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

f the most elegant derivative results and is closely related to the integral ∫ 1/x dx = ln|x| + C.

自然对数 ln x（x > 0）的导数是 1/x。这是最优美的导数结果之一，并与积分 ∫ 1/x dx = ln|x| + C 密切相关。

d/dx (ln x) = 1/x

For logarithms with an arbitrary positive base a ≠ 1, the derivative is d/dx (log_a x) = 1/(x ln a). This follows from the change-of-base formula log_a x = (ln x)/(ln a).

对于任意正底数 a ≠ 1 的对数，其导数为 d/dx (log_a x) = 1/(x ln a)。这可以通过换底公式 log_a x = (ln x)/(ln a) 得到。

d/dx (log_a x) = 1/(x ln a)

Example: d/dx (log₂ common logarithm (base 10) gives 1/(x ln 10). In all these cases, the domain is x > 0.

例子：d/dx (log₂ x) = 1/(x ln 2)；常用对数（底为 10）的导数 d/dx (log x) 为 1/(x ln 10)。在所有这些情况下，定义域均为 x > 0。

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

x. From these, the others can be derived using the quotient rule or reciprocal identities.

六个基本三角函数的导数是重要的基石。最核心的两个结果是 d/dx (sin x) = cos x 和 d/dx (cos x) = −sin x。由它们出发，利用商法则或倒数关系就能推导出其他三角函数的导数。

d/dx (sin x) = cos x    |    d/dx (cos x) = −sin x

For the tangent function, tan x = sin x / cos x, so d/dx (tan x) = sec² x. Equivalently, this can be written as 1/(cos² x). Note that sec x = 1/cos x, and its derivative is sec x tan x.

对于正切函数 tan x = sin x / cos x，因此 d/dx (tan x) = sec² x。这也可以等价地写为 1/(cos² x)。注意 sec x = 1/cos x，其导数为 sec x tan x。

d/dx (tan x) = sec² x    |    d/dx (sec x) = sec x tan x

ining three derivatives complete the set: d/dx (csc x) = −csc x cot x, d/dx (cot x) = −csc² x. Here csc x = 1/sin x and cot x = cos x / sin x.

剩下的三个导数完整了这张表：d/dx (csc x) = −csc x cot x，d/dx (cot x) = −csc² x。其中 csc x = 1/sin x，cot x = cos x / sin x。

d/dx (csc x) = −csc x cot x    |    d/dx (cot x) = −csc² x

Notice the symmetry: the derivatives of co-functions (cos, csc, cot) all carry a minus sign, whereas the derivatives of sin and tan do not. This pattern can help with memorisation.

请注意其中的对称性：余函数（cos、csc、cot）的导数都带有

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