📚 Mastering Differentiation: Exercise 12A | 掌握微分法：练习 12A
Differentiation is a cornerstone of IB Mathematics, and Exercise 12A typically introduces the fundamental rules for finding derivatives. This article guides you through the key concepts you will encounter in such an exercise, from the power rule to the quotient rule, all illustrated with worked examples. Understanding these rules deeply will not only help you solve Exercise 12A but also build a solid foundation for calculus throughout the course.
微分法是 IB 数学的基石，练习 12A 通常会介绍求导的基本法则。本文将带你逐一梳理这类练习中涉及的核心概念，从幂法则到商法则，全部辅以详细例题。深入理解这些法则不仅能帮助你顺利解答练习 12A，更将为整个课程中的微积分学习打下坚实基础。
1. Understanding the Power Rule | 理解幂法则
If f(x) = xⁿ, where n is any real number, then its derivative is f' entiation, allowing you to handle any simple power of x. For example, if f(x) = x⁵, then f'(x) = 5x⁴. If f(x) = √x, first rewrite as x¹⁄², then f'(x) = (1/2)x⁻¹⁄² = 1/(2√x).
若 f(x) = xⁿ，其中 n 为任意实数，则其导数为 f'(x) = n xⁿ⁻¹。这是求导中最常用的法则，能处理任何简单的 x 的幂次。例如，若 f(x) = x⁵，则 f'(x) = 5x⁴。若 f(x) = √x，先改写为 x¹⁄²，则 f'(x) = (1/2)x⁻¹⁄² = 1/(2√x)。
Power Rule: d/dx (xⁿ) = n xⁿ⁻¹
2. Constant Multiple Rule | 常数倍法则
constant multiplied by the derivative of the function. In symbols, d/dx [c · f(x)] = c · f'(x). For instance, if f(x) = 3x⁴, then f'(x) = 3 · 4x³ = 12x³.
当函数乘以一个常数时，导数等于该常数乘以函数的导数。用符号表示为 d/dx [c · f(x)] = c · f'(x)。例如，若 f(x) = 3x⁴，则 f'(x) = 3 · 4x³ = 12x³。
d/dx [c f(x)] = c f'(x)
3. Sum and Difference Rules | 和差法则
e sum (or difference) of the derivatives. So if you have f(x) = u(x) + v(x) - w(x), then f'(x) = u'(x) + v'(x) - w'(x). Example: f(x) = 2x³ - 5x² + 7x - 1 gives f'(x) = 6x² - 10x + 7.
和（或差）的导数等于导数的和（或差）。因此，若 f(x) = u(x) + v(x) - w(x)，则 f'(x) = u'(x) + v'(x) - w'(x)。例如，f(x) = 2x³ - 5x² + 7x - 1 的导数为 f'(x) = 6x² - 10x + 7。
These three basic rules are often combined in Exercise 12A. Break down complicated polynomials into individual terms and differentiate term by term.
这三个基本法则经常在练习 12A 中组合使用。将复杂的多项式拆分成单项，然后逐项求导即可。
4. Product Rule | 乘积法则
differentiate a product of two functions, y = u(x) · v(x), use the product rule: dy/dx = u' v + u v'. A typical IB example: Differentiate f(x) = x² · eˣ. Here u = x², v = eˣ, so u' = 2x, v' = eˣ. Hence f'(x) = 2x eˣ + x² eˣ = x eˣ (2 + x).
当你需要求两个函数乘积的导数时，即 y = u(x) · v(x)，使用乘积法则：dy/dx = u' v + u v'。一个典型的 IB 例子：对 f(x) = x² · eˣ 求导。这里 u = x², v = eˣ，因此 u' = 2x，v' = eˣ。于是 f'(x) = 2x eˣ + x² eˣ = x eˣ (2 + x)。
Memorise the pattern: "derivative of first times second, plus first times derivative of second." Many Exercise 12A problems test this rule, especially with polynomials and trigonometric functions combined.
记住这个模式：“第一个的导数乘第二个，加上第一个乘第二个的导数。”练习 12A 中有很多题目考察此法则，尤其是多项式与三角函数的组合。
5. Quotient Rule | 商法则
For a quotient y = u/v, the derivative is given by (v u' - u v' v². For example, differentiate f(x) = (x² + 1) / (x - 2). Set u = x² + 1, v = x - 2, so u' = 2x, v' = 1. Then f' 2x) - (x² + 1)(1)] / (x - 2)² = (2x² - 4x - x² - 1) / (x - 2)² = (x² - 4x - 1) / (x - 2)².
对于分式 y = u/v，其导数公式为 (v u' - u v') / v²。例如，对 f(x) = (x² + 1) / (x - 2) 求导。令 u = x² + 1，v = x - 2，则 u' = 2x，v' = 1。于是 f' (2x) - (x² + 1)(1)] / (x - 2)² = (2x² - 4x - x² - 1) / (x - 2)² = (x² - 4x - 1) / (x - 2)²。
A common mistake is to reverse the subtraction order in the numerator. Always remember: "low d-high minus high d-low, over the square of what's below."
一个常见错误是分子中的减法顺序颠倒。请始终记住：“下乘上导，减去上乘下导，除以下面的平方。”
6. Chain Rule (Basic) | 链式法则（基础）
The chain rule is used for composite functions. If y = f[g(x)], then dy/dx = f'[g(x)] · g' Here outer function is power 4, inner is 3x - 1. So dy/dx = 4(3x - 1)³ · 3 = 12(3x - 1)³.
链式法则用于复合函数。若 y = f[g(x)]，则 dy/dx = f'[g(x)] · g'(x)。在 IB 练习 12A 中，你会看到简单的应用，如 y = (3x - 1)⁴。这里外层函数是四次幂，内层是 3x - 1。因此 dy/dx = 4(3x - 1)³ · 3 = 12(3x - 1)³。
Another common type: y = √2x + 5 = (2x + 5)¹⁄². Then f'(x) = (1/2)(2x + 5)⁻¹⁄² · 2 = 1/√2x + 5.
另一种常见类型：y = √2x + 5 = (2x + 5)¹⁄²。则 f'(x) = (1/2)(2x + 5)⁻¹⁄² · 2 = 1/√2x + 5。
7. Differentiating Exponential and Logarithmic Functions | 指数与对数函数求导
For the natural exponential function: d/dx (eˣ) = eˣ. More generally, d/dx (eᵏˣ) = k eᵏˣ hms: d/dx (ln x) = 1/x for x > 0. If you have ln(f(x)), use the chain rule: d/dx ln(u) = u' / u.
对于自然指数函数：d/dx (eˣ) = eˣ。更一般地，d/dx (eᵏˣ) = k eᵏˣ。对于自然对数：d/dx (ln x) = 1/x，其中 x > 0。若涉及 ln(f(x))，使用链式法则：d/dx ln(u) = u' / u。
Exercise 12A often includes problems like differentiating e³ˣ or ln(5x). Practise these as they form the basis for more complicated calculus later.
练习 12A 经常包含对 e³ˣ 或 ln(5x) 求导的题目。多做练习，因为这些是后续更复杂微积分的基础。
8. Trigonometric Functions Differentiation | 三角函数求导
s x) = -sin x, d/dx (tan x) = sec² x. With chain rule, d/dx (sin(ax + b)) = a cos(ax + b).
IB 公式手册会提供这些公式，但为了高效解题，你必须牢记：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec² x。结合链式法则，d/dx (sin(ax + b)) = a cos(ax + b)。
In Exercise 12A, you may be asked to differentiate expressions like 2 sin(3x) + cos(2x). The derivative would be 6 cos(3x) - 2 sin(2x).
在练习 12A 中，你可能会被要求对诸如 2 sin(3x) + cos(2x) 的表达式求导。其导数应为 6 cos(3x) - 2 sin(2x)。
9. Higher-Order Derivatives | 高阶导数
The second derivative, f'' mply the derivative of the first derivative. In Exercise 12A, you might be asked to find f''(x) after finding f'(x). For example, if f(x) = x⁴ - 2x³ + x, then f'(x) = 4x³ - 6x² + 1, and f'' ² - 12x. Higher-order derivatives are used for concavity and optimisation in later topics.
二阶导数 f''(x) 或 d²y/dx² 就是一阶导数的导数。在练习 12A 中，你可能会被要求在求出 f'(x) 后再求 f''(x)。例如，若 f(x) = x⁴ - 2x³ + x，则 f'(x) = 4x³ - 6x² + 1，f''(x) = 12x² - 12x。高阶导数将在后续主题中用于判断凹凸性和优化问题。
Pay attention to notation: f'(x), dy/dx, d/dx all mean first derivative; f''(x), d²y/dx² mean second derivative.
注意符号：f'(x)、dy/dx、d/dx 均表示一阶导数；f''(x)、d²y/dx² 表示二阶导数。
10. Applications: Tangents and Normals | 应用：切线与法线
asks for the equation of a tangent or normal at a point. The gradient of the tangent is f'(a) at x = a. The tangent line passes through (a, f(a)): y - f(a) = f'(a)(x - a). The normal is perpendicular: gradient = -1 / f'(a), provided f'(a) ≠ 0.
练习 12A 常常要求求某一点处的切线或法线方程。切线的斜率是在 x = a 处的 f'(a)。切线经过点 (a, f(a))：y - f(a) = f'(a)(x - a)。法线与之垂直，斜率为 -1 / f'(a)，前提是 f'(a) ≠ 0。
Example: Find the tangent to y = x² + 3 at x = 1. f'(x) = 2x, so m = 2. Point is (1,4). Tangent: y - 4 = 2(x - 1) → y = 2x + 2.
示例：求曲线 y = x² + 3 在 x = 1 处的切线。f'(x) = 2x，故斜率 m = 2。点为 (1,4)。切线方程为 y - 4 = 2(x - 1) → y = 2x + 2。
This application ties differentiation to coordinate geometry and is a classic IB problem type.
这一应用将微分法与坐标几何联系起来，是典型的 IB 题型。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图