IB Mathematics: Fundamental Concepts and Rules of Differentiation | IB数学：微分基本概念与运算法则

📚 IB Mathematics: Fundamental Concepts and Rules of Differentiation | IB数学：微分基本概念与运算法则

Differentiation is one of the cornerstones of calculus and a core topic in IB Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI). This article introduces the fundamental concepts of derivatives and the essential rules for differentiating functions, following the IB syllabus requirements.

微分是微积分的基石之一，也是IB数学分析与方法（AA）以及应用与解释（AI）中的核心内容。本文将按照IB考纲要求，介绍导数的基本概念以及求导的基本运算法则。

1. The Concept of the Derivative | 导数的概念

The derivative of a function f at a point x, denoted f'(x), measures the instantaneous rate of change of f with respect to x. Geometrically, it is the slope of the tangent line to the curve y = f(x) at that point.

函数 f 在点 x 处的导数，记作 f'(x)，衡量的是 f 关于 x 的瞬时变化率。从几何上看，它是曲线 y = f(x) 在该点处切线的斜率。

If the limit exists, we define the derivative as:

若下列极限存在，则导数定义为：

f'(x) = lim_h→0 [f(x + h) − f(x)] / h

This definition is sometimes called the first principle of differentiation. In IB, you may be asked to differentiate simple functions from first principles, especially for linear and quadratic functions.

这一定义有时被称为微分的第一性原理。在IB考试中，你可能会被要求利用第一性原理对简单函数（尤其是线性函数和二次函数）求导。

2. Notation for Derivatives | 导数的表示方法

There are several standard notations for derivatives. You should be comfortable with all of them, as different contexts may call for different notation.

导数有几种标准记号。你应该熟悉所有这些记号，因为不同情境下可能会使用不同的表示方式。

Leibniz notation: dy/dx, d/dx [f(x)]

莱布尼茨记号：dy/dx，d/dx [f(x)]
Lagrange notation: f'(x), y’

拉格朗日记号：f'(x)，y’
Newton notation (mostly for time): ẏ, ḟ

牛顿记号（多用于对时间）：ẏ，ḟ
Euler notation: Df(x)

欧拉记号：Df(x)

In IB exams, Leibniz notation is often preferred for applications such as related rates and kinematics, while Lagrange notation appears in algebraic manipulation.

在IB考试中，莱布尼茨记号常用于相关变化率和运动学等应用，而拉格朗日记号则常见于代数运算中。

3. Differentiability and Continuity | 可导性与连续性

A function is differentiable at a point if its derivative exists there. For a function to be differentiable at x = a, it must be continuous at a, but continuity alone does not guarantee differentiability.

一个函数在某点可导，意味着该点的导数存在。函数在 x = a 处可导，则它必在 a 处连续，但连续并不能保证可导。

Common cases where a function is continuous but not differentiable include:

函数连续但不可导的常见情形包括：

Corner points (e.g., f(x) = |x| at x = 0).

尖点（例如 f(x) = |x| 在 x = 0 处）。
Vertical tangents (infinite slope).

垂直线切线（斜率无穷大）。
Discontinuities (though then it is not continuous either).

间断点（这种情况当然也不连续）。

Remember: differentiability ⇒ continuity, but the converse is false.

请记住：可导 ⇒ 连续，但逆命题不成立。

4. The Power Rule | 幂函数法则

The power rule states that for any real number n,

幂函数法则指出，对于任意实数 n，

d/dx [xⁿ] = n xⁿ⁻¹

This is one of the most frequently used rules. It applies to negative and fractional exponents too, as long as the function is defined for the given x.

这是最常用的法则之一。它对负指数和分数指数同样适用，只要函数在给定的 x 处有定义。

Example: f(x) = x⁵ ⇒ f'(x) = 5x⁴.

例：f(x) = x⁵ ⇒ f'(x) = 5x⁴。

Example: f(x) = √x = x^1/2 ⇒ f'(x) = ½ x^−1/2 = 1/(2√x).

例：f(x) = √x = x^1/2 ⇒ f'(x) = ½ x^−1/2 = 1/(2√x)。

5. The Constant Multiple and Sum Rules | 常数倍法则与和法则

The derivative of a constant times a function is the constant times the derivative of the function:

常数乘以函数的导数，等于常数乘以该函数的导数：

d/dx [c f(x)] = c f'(x)

The derivative of a sum is the sum of the derivatives:

和的导数等于导数的和：

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Together these allow us to differentiate any polynomial term by term.

这两个法则结合使用，我们可以对多项式逐项求导。

Example: d/dx [3x² + 2x − 1] = 6x + 2.

例：d/dx [3x² + 2x − 1] = 6x + 2。

6. The Product Rule | 乘积法则

If u = f(x) and v = g(x), then the derivative of the product uv is given by:

若 u = f(x)、v = g(x)，则乘积 uv 的导数由下式给出：

d/dx [u v] = u (dv/dx) + v (du/dx)

A common mnemonic is “first times derivative of second plus second times derivative of first.” Do not simply differentiate each factor separately.

常用口诀是“前导后不导加上后导前不导”。切勿将每个因子分别求导后相乘。

Example: f(x) = x² sin x. Let u = x², v = sin x.

例：f(x) = x² sin x。设 u = x²，v = sin x。

f'(x) = x² cos x + 2x sin x

7. The Quotient Rule | 商法则

If u and v are differentiable functions of x, and v ≠ 0, then:

若 u 和 v 都是关于 x 的可导函数，且 v ≠ 0，则：

d/dx [u/v] = (v (du/dx) − u (dv/dx)) / v²

The order of the numerator matters: it is “low d high minus high d low over low squared.”

注意分子的顺序：口诀为“下乘上导减上乘下导，除以分母的平方”。

Example: f(x) = (x + 1)/(x − 1). Using u = x + 1, v = x − 1:

例：f(x) = (x + 1)/(x − 1)。令 u = x + 1，v = x − 1：

f'(x) = [(x − 1)(1) − (x + 1)(1)] / (x − 1)² = −2 / (x − 1)²

8. The Chain Rule | 链式法则

The chain rule is used to differentiate composite functions. If y = f(u) and u = g(x), then:

链式法则用于求复合函数的导数。若 y = f(u)，u = g(x)，则：

dy/dx = (dy/du) × (du/dx)

Equivalently, if f(x) = h(g(x)), then f'(x) = h'(g(x)) g'(x).

等价地，若 f(x) = h(g(x))，则 f'(x) = h'(g(x)) g'(x)。

Example: y = (3x² + 1)⁴. Let u = 3x² + 1, so y = u⁴.

例：y = (3x² + 1)⁴。令 u = 3x² + 1，则 y = u⁴。

dy/dx = 4u³ × 6x = 24x(3x² + 1)³

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

The derivatives of the six basic trigonometric functions must be memorized. In IB, angles are always measured in radians unless otherwise stated.

六个基本三角函数的导数必须牢记。在IB中，如无特别说明，角度均以弧度制计量。

f(x)	f'(x)	定义域说明
sin x	cos x	所有实数
cos x	−sin x	所有实数
tan x	sec² x	x ≠ π/2 + kπ
csc x	−csc x cot x	x ≠ kπ
sec x	sec x tan x	x ≠ π/2 + kπ
cot x	−csc² x	x ≠ kπ

Remember that these derivatives require the chain rule when the argument is not simply x. For example, d/dx [sin(2x)] = 2 cos(2x).

请注意，当自变量不是简单的 x 而是复合函数时，需要使用链式法则。例如，d/dx [sin(2x)] = 2 cos(2x)。

10. Derivatives of Exponential and Logarithmic Functions | 指数函数与对数函数的导数

The natural exponential function has the remarkable property that it is its own derivative:

自然指数函数有一个重要性质：它的导数等于它本身：

d/dx [eˣ] = eˣ

For a general base a > 0, a ≠ 1:

对于一般底数 a > 0，a ≠ 1：

d/dx [aˣ] = aˣ ln a

The natural logarithm has the derivative:

自然对数的导数为：

d/dx [ln x] = 1/x, x > 0

For general base a:

对于一般底数 a：

d/dx [logₐ x] = 1/(x ln a)

Using the chain rule, d/dx [e^kx] = k e^kx, and d/dx [ln(kx)] = 1/x for any constant k ≠ 0.

利用链式法则，d/dx [e^kx] = k e^kx，且 d/dx [ln(kx)] = 1/x（对任意常数 k ≠ 0）。

11. Implicit Differentiation | 隐函数求导

When a curve is given by an equation that cannot be easily solved for y explicitly, we can differentiate both sides with respect to x, treating y as a differentiable function of x. Each time we differentiate a term involving y, we multiply by dy/dx.

当曲线方程不易显式解出 y 时，我们可以对等式两边关于 x 求导，将 y 视为关于 x 的可导函数。每次对含 y 的项求导后，都要乘以 dy/dx。

Example: x² + y² = 25.

例：x² + y² = 25。

2x + 2y (dy/dx) = 0 ⇒ dy/dx = −x/y

Implicit differentiation is essential for finding tangent lines to curves not representable by a single explicit function.

隐函数求导对于求那些无法用单一显函数表示的曲线的切线至关重要。

12. Higher-Order Derivatives and Applications | 高阶导数及应用

The second derivative f”(x) is the derivative of f'(x). It represents the rate of change of the slope, which relates to concavity. Higher-order derivatives appear in Taylor expansions and kinematics (acceleration is the second derivative of displacement).

二阶导数 f”(x) 是一阶导数 f'(x) 的导数。它表示斜率的变化率，与函数的凹凸性有关。更高阶导数出现在泰勒展开和运动学中（加速度是位移的二阶导数）。

Common applications of differentiation in IB include:

IB中导数的常见应用包括：

Finding equations of tangents and normals.

求切线与法线方程。
Solving optimization problems.

求解最优化问题。
Determining increasing/decreasing intervals.

判断函数的增减区间。
Finding stationary points and points of inflection.

求驻点和拐点。
Kinematics: velocity and acceleration.

运动学中的速度与加速度。

To master differentiation, practice applying these rules in combination, and always check the domain and differentiability conditions.

要掌握微分，请练习综合运用这些法则，并始终检查定义域和可导性条件。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Mathematics: Fundamental Concepts and Rules of Differentiation | IB数学：微分基本概念与运算法则