IB & CCEA Maths: Differentiation – Key Concepts | IB CCEA 数学：微分考点精讲

📚 IB & CCEA Maths: Differentiation – Key Concepts | IB CCEA 数学：微分考点精讲

Differentiation is a cornerstone of calculus, appearing in both IB and CCEA Mathematics. It allows us to determine how a function changes at any given point, giving the gradient of a curve and underpinning applications like optimisation and modelling. This revision guide breaks down the key concepts, rules, and applications you need to master.

微分是微积分的重要基石，在 IB 与 CCEA 数学中均占有核心地位。它帮助我们了解函数在任意一点的变化情况，给出曲线的斜率，并支撑着最优化、建模等应用。本考点精讲将逐一解析你需掌握的关键概念、求导法则及应用。

1. Introduction to Differentiation | 微分简介

The derivative of a function f(x) with respect to x is written as f'(x) or dy/dx. It represents the instantaneous rate of change of the dependent variable y with respect to the independent variable x. In simple terms, it answers the question: how fast is y changing as x changes?

函数 f(x) 关于 x 的导数记作 f'(x) 或 dy/dx，它表示因变量 y 相对于自变量 x 的瞬时变化率。简单来说，它回答了这样一个问题：当 x 变化时，y 的变化有多快？

Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point. If the function is a straight line, the derivative is constant; if it is a curve, the derivative varies along the curve.

从几何角度看，某点的导数就是该点处函数图像切线的斜率。如果函数是直线，导数恒定；如果是曲线，导数则会沿着曲线变化。

2. Limits and the Definition of Derivative | 极限与导数定义

The formal definition of the derivative relies on the concept of a limit:

导数的严格定义依赖于极限的概念：

f'(x) = lim (h→0) [f(x+h) – f(x)] / h

This expression represents the limit of the average rate of change as the interval shrinks to zero. If the limit exists, we say the function is differentiable at that point.

该表达式表示当区间缩小至零时平均变化率的极限。若极限存在，则称函数在该点可导。

For example, to differentiate f(x)=x² from first principles, substitute into the definition:

例如，用第一原理求 f(x)=x² 的导数，代入定义可得：

lim (h→0) [(x+h)² – x²] / h = lim (h→0) [2xh + h²] / h = 2x

Thus f'(x)=2x. A function that is not continuous at a point cannot be differentiable there.

因此 f'(x)=2x。在一点不连续的函数在该点必定不可导。

3. Basic Differentiation Rules | 基本求导法则

These fundamental rules let you differentiate polynomials and combinations of functions quickly without returning to the limit definition each time.

运用下列基本法则，无需每次都回到极限定义，就能快速对多项式及函数组合求导。

Constant Rule: d/dx (c) = 0, where c is a constant.
Power Rule: d/dx (xⁿ) = n xⁿ⁻¹.
Constant Multiple Rule: d/dx [c f(x)] = c f'(x).
Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

常数法则：d/dx (c) = 0，c 为常数。
幂法则：d/dx (xⁿ) = n xⁿ⁻¹。
常数倍法则：d/dx [c f(x)] = c f'(x)。
和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

Example: If f(x) = 4x³ – 2x + 7, then f'(x) = 12x² – 2.

示例：若 f(x) = 4x³ – 2x + 7，则 f'(x) = 12x² – 2。

4. The Chain Rule | 链式法则

The chain rule is used when differentiating a composite function, i.e. a function inside another function. If y = f(g(x)), then:

链式法则用于求复合函数的导数，即函数内部还有函数。若 y = f(g(x))，则：

dy/dx = f'(g(x)) · g'(x)

In words, differentiate the outer function, leave the inner function untouched, then multiply by the derivative of the inner function.

用语言表述：先对外层函数求导，内层函数暂时保留，再乘以内层函数的导数。

Example: y = sin(5x). Outer: sin u → cos u; inner: u=5x → 5. So dy/dx = cos(5x) · 5 = 5cos(5x).

示例：y = sin(5x)。外层：sin u → cos u；内层：u=5x → 5。因此 dy/dx = cos(5x) · 5 = 5cos(5x)。

For more complex expressions like y = (3x²+1)⁴, set u=3x²+1, then dy/dx = 4u³ · 6x = 24x(3x²+1)³.

对于 y = (3x²+1)⁴ 等更复杂的表达式，令 u=3x²+1，则 dy/dx = 4u³ · 6x = 24x(3x²+1)³。

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

When two functions are multiplied or divided, you need special rules.

当两个函数相乘或相除时，需要使用专门的法则。

Product Rule: If y = u(x)·v(x), then dy/dx = u’v + uv’.

积法则：若 y = u(x)·v(x)，则 dy/dx = u’v + uv’。

Example: y = x²·sin x. Let u=x² (u’=2x) and v=sin x (v’=cos x). Then dy/dx = 2x·sin x + x²·cos x.

示例：y = x²·sin x。设 u=x² (u’=2x)，v=sin x (v’=cos x)，则 dy/dx = 2x·sin x + x²·cos x。

Quotient Rule: If y = u(x)/v(x), then dy/dx = (u’v – uv’) / v².

商法则：若 y = u(x)/v(x)，则 dy/dx = (u’v – uv’) / v²。

Example: y = x / (x+1). u=x, v=x+1. u’=1, v’=1. Then dy/dx = [1·(x+1) – x·1] / (x+1)² = 1/(x+1)².

示例：y = x / (x+1)。u=x，v=x+1，u’=1，v’=1。则 dy/dx = [1·(x+1) – x·1] / (x+1)² = 1/(x+1)²。

6. Derivatives of Trigonometric, Exponential, and Logarithmic Functions | 三角函数、指数与对数函数的导数

You must memorise the derivatives of these standard functions. They form the building blocks for more complicated differentiation problems.

你必须牢记以下标准函数的导数，它们是解决更复杂求导问题的基础。

f(x)	f'(x)
xⁿ	n xⁿ⁻¹
eˣ	eˣ
aˣ	aˣ ln a
ln x	1/x
sin x	cos x
cos x	–sin x
tan x	sec² x

When these functions are combined with the chain rule, remember to multiply by the derivative of the inner expression. For example, d/dx (e²ˣ) = 2e²ˣ, and d/dx (ln(3x)) = 1/x.

当这些函数与链式法则结合时，务必乘以内层表达式的导数。例如，d/dx (e²ˣ) = 2e²ˣ，d/dx (ln(3x)) = 1/x。

7. Implicit Differentiation | 隐函数求导

Sometimes y is not given explicitly as a function of x, for instance in equations like x² + y² = 25. To find dy/dx, differentiate both sides of the equation with respect to x, treating y as a function of x and using the chain rule for y terms.

有时 y 并未显式表示为 x 的函数，例如在方程 x² + y² = 25 中。为求 dy/dx，需对等式两边关于 x 求导，将 y 视为 x 的函数，并对包含 y 的项使用链式法则。

Differentiating x² + y² = 25: 2x + 2y (dy/dx) = 0, so dy/dx = –x/y. Notice the derivative appears again inside the solution.

对 x² + y² = 25 求导：2x + 2y (dy/dx) = 0，因此 dy/dx = –x/y。注意导数出现在解的表达式中。

For more involved expressions like eʸ + xy = 1, you differentiate term by term, then collect all dy/dx terms on one side and factorise to solve for dy/dx.

对于 eʸ + xy = 1 等更复杂的表达式，逐项求导后，将所有含 dy/dx 的项移到一边，通过因式分解解出 dy/dx。

8. Higher-Order Derivatives | 高阶导数

The derivative of a derivative is called the second derivative, written as f”(x) or d²y/dx². It measures the rate of change of the slope, i.e. the concavity of the graph. In kinematics, if s(t) is displacement, then s'(t) is velocity and s”(t) is acceleration.

导数的导数称为二阶导数，记作 f”(x) 或 d²y/dx²。它衡量斜率的变化率，即函数图像的凹凸性。在运动学中，若 s(t) 表示位移，则 s'(t) 为速度，s”(t) 为加速度。

For a function f(x), if f”(x) > 0 on an interval, the graph is concave up (shaped like a cup); if f”(x) < 0, it is concave down. Points where f''(x) = 0 or changes sign may be inflection points.

对于函数 f(x)，若在某一区间内 f”(x) > 0，图像是凹向上的（呈杯形）；若 f”(x) < 0，则为凹向下。f''(x) = 0 或变号的点可能是拐点。

9. Applications: Tangents, Normals, and Rates of Change | 应用：切线、法线与变化率

The derivative gives the slope of the tangent at a point (x₁, y₁): m = f'(x₁). The equation of the tangent line is y – y₁ = m(x – x₁). The normal line is perpendicular to the tangent, so its slope is –1/m and its equation is y – y₁ = (–1/m)(x – x₁).

导数给出点 (x₁, y₁) 处切线的斜率：m = f'(x₁)。切线方程为 y – y₁ = m(x – x₁)。法线垂直于切线，因此其斜率为 –1/m，方程为 y – y₁ = (–1/m)(x – x₁)。

Rates of change are direct applications of derivatives. For example, if the radius r of a circle increases at a constant rate dr/dt, then the rate of change of the area A is dA/dt = 2πr (dr/dt).

变化率是导数的直接应用。例如，若圆的半径 r 以恒定速率 dr/dt 增大，则面积 A 的变化率为 dA/dt = 2πr (dr/dt)。

10. Stationary Points and Curve Sketching | 驻点与曲线草图

Stationary points occur where f'(x) = 0. These can be local maxima, local minima, or points of inflection with a horizontal tangent. To classify them, use either the first derivative test (sign change of f’) or the second derivative test.

驻点出现在 f'(x) = 0 处。它们可能是局部极大值点、局部极小值点或具有水平切线的拐点。分类时常使用一阶导数符号检验法或二阶导数检验法。

Second derivative test: If f”(a) > 0, then x=a is a local minimum; if f”(a) < 0, it is a local maximum. If f''(a) = 0, the test is inconclusive and you should examine the sign of f' either side of a.

二阶导数检验：若 f”(a) > 0，则 x=a 为局部极小点；若 f”(a) < 0，则为局部极大点。若 f''(a) = 0，检验失效，需查看 a 两侧 f' 的符号。

For curve sketching, combine derivatives to find intercepts, stationary points, concavity, and asymptotes to produce an accurate graph.

在绘制曲线草图时，应综合利用导数求出截距、驻点、凹凸性以及渐近线，以绘制准确的图形。

11. Optimization Problems | 最优化问题

Optimisation involves finding maximum or minimum values of a quantity, a frequent requirement in both IB and CCEA exams. The steps are:

最优化问题要求找出某个量的最大值或最小值，在 IB 与 CCEA 考试中十分常见。基本步骤如下：

Express the quantity to be optimised as a function of one variable, using given constraints.
Differentiate the function to find f'(x).
Set f'(x) = 0 to locate stationary points.
Use the second derivative test or a sign table to confirm the nature of the stationary points.
Check endpoints if the domain is restricted, as the absolute maximum/minimum may occur there.

利用给定约束，将待优化量表示为单一变量的函数。
对函数求导，得到 f'(x)。
令 f'(x) = 0，找到驻点。
运用二阶导数检验或符号表，确认驻点的性质。
若定义域有限，需检查端点，因为绝对最大值或最小值可能出现在端点处。

A classic example: find the dimensions of a rectangle with fixed perimeter that maximise the area. Let x be width, then length = (P/2 – x), area A = x(P/2 – x). Differentiate, set A’=0 and solve.

经典例题：在周长固定的情况下，求使面积最大的矩形尺寸。设宽为 x，则长为 (P/2 – x)，面积 A = x(P/2 – x)。求导、令 A’=0 并求解即可。

12. Related Rates | 相关变化率

Related rates problems involve two or more quantities that vary with time, linked by an equation. You differentiate the entire equation with respect to time t, applying the chain rule implicitly, to relate their rates of change.

相关变化率问题涉及两个或多个随时间变化的量，它们由一个方程联系在一起。需将整个方程对时间 t 求导，隐式地运用链式法则，从而建立变化率之间的关系。

For example, a spherical balloon is being inflated so that its volume increases at 100 cm³/s. To find how fast the radius increases when r=5 cm, start with V = (4/3)πr³. Differentiate both sides with respect to t: dV/dt = 4πr² (dr/dt). Substitute dV/dt=100 and r=5 to solve for dr/dt.

例如，一个球形气球以 100 cm³/s 的速率膨胀。求当半径 r=5 cm 时半径的增加速率。由 V = (4/3)πr³ 入手，两边对 t 求导：dV/dt = 4πr² (dr/dt)。代入 dV/dt=100 和 r=5，即可解出 dr/dt。

Key tip: always identify the given rate, the required rate, and an equation linking the variables before differentiating. Be careful to substitute values only after differentiation.

关键技巧：在求导之前，务必明确已知速率、所求速率以及变量间的联系方程。务必注意，求导后才可以代入具体数值。

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IB & CCEA Maths: Differentiation – Key Concepts | IB CCEA 数学：微分考点精讲