📚 Exercise 8A | 练习8A
Exercise 8A in the IB Mathematics curriculum typically introduces the basic techniques of differentiation. It lays the groundwork for understanding how functions change and how to compute derivatives efficiently using standard rules. This revision guide walks you through every core concept covered in this exercise, complete with worked examples and key insights to help you master differentiation from first principles to polynomial derivatives and beyond.
IB 数学课程中的练习 8A 一般会介绍基本的微分技巧。它为你理解函数如何变化以及如何利用标准法则快速计算导数打下了基础。这篇复习指南将带你逐一梳理该练习涉及的每一个核心概念，配有详细的示例和关键要点，帮助你掌握从第一性原理到多项式导数及更深层次的内容。
1. What is Differentiation? | 什么是微分？
Differentiation is a branch of calculus that deals with the rate at which a quantity changes. In graphical terms, the derivative of a function at a point gives the gradient of the tangent line to the curve at that point. It allows us to analyse motion, growth, optimisation and many other real-world phenomena.
微分是微积分的一个分支，研究量的变化率。从图形上看，函数在某一点的导数给出了该点处曲线切线的斜率。它使我们能够分析运动、增长、最优化以及许多其他现实世界中的现象。
Consider a function y = f(x). The derivative, written as f' means the function is increasing, while a negative derivative indicates it is decreasing.
考虑函数 y = f(x)。导数记作 f'(x) 或 dy/dx，它告诉我们函数图像在任意 x 值处的陡峭程度。导数为正意味着函数递增，导数为负则表示函数递减。
In IB Mathematics, you are expected to move quickly from the idea of a limit to applying differentiation rules. Exercise 8A ensures you are confident with the fundamental building blocks.
在 IB 数学中，你需要迅速从极限的概念过渡到应用微分法则。练习 8A 就是为了确保你能扎实掌握这些基本模块。
2. Gradient of a Curve and First Principles | 曲线的斜率与第一性原理
The derivative can be defined rigorously using the idea of a limit. The gradient of a chord joining two points on the curve approaches the gradient of the tangent as the two points come infinitely close. This is the first principles definition:
导数可以通过极限的概念严格定义。曲线上的两点连成的弦的斜率，在两点无限接近时趋近于切线的斜率。这就是第一性原理的定义：
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
8A, you often start by using this limit to differentiate simple functions like f(x) = x².
这里的 h 表示 x 的一个微小变化量。当 h 趋向于零时，这个商就给出了瞬时变化率。在练习 8A 中，你通常会从使用这个极限来对 f(x) = x² 这样的简单函数求导开始。
e difference quotient simplifies to (2xh + h²)/h = 2x + h. Taking the limit as h→0 gives f'(x) = 2x. This process reinforces the meaning of the derivative.
对于 f(x) = x²，展开式得到 f(x+h) = (x+h)² = x² + 2xh + h²。然后差商化简为 (2xh + h²)/h = 2x + h。取 h→0 时的极限就得到 f'(x) = 2x。这一过程强化了导数的含义。
While you will not be required to use first principles for every differentiation in an exam, understanding it is crucial for grasping why the power rule works. Exercise 8A often includes a few problems that ask you to derive from first principles to cement your understanding.
尽管考试中不会要求你对每次微分都使用第一性原理，但理解它对于掌握幂法则为什么成立至关重要。练习 8A 通常会包含一些要求你用第一性原理推导的题目，以巩固你的理解。
3. The Power Rule | 幂法则
ifferentiation shortcut. If f(x) = xⁿ, where n is any real number, then the derivative is:
幂法则是使用最频繁的微分速算方法。如果 f(x) = xⁿ，其中 n 是任意实数，那么导数为：
d/dx (xⁿ) = n xⁿ⁻¹
since x = x¹ and 1·x⁰ = 1. This rule makes differentiating monomials almost instantaneous.
例如，x³ 的导数是 3x²；x⁵ 的导数是 5x⁴；而 x 的导数就是 1，因为 x = x¹，而 1·x⁰ = 1。这个法则让单项式的微分几乎瞬间完成。
Exercise 8A will give you plenty of practice with positive integer powers, but the power rule also holds for negative and fractional powers. For instance, differentiating x⁻² yields -2x⁻³, and differentiating √x = x1/2 gives (1/2)x⁻1/2.
练习 8A 将提供大量正整数指数幂的练习，但幂法则同样适用于负指数和分数指数。例如，对 x⁻² 求导得到 -2x⁻³；而对 √x = x1/2 求导则得到 (1/2)x⁻1/2。
Always rewrite radicals and reciprocals as powers before differentiating. This avoids errors and allows you to stick to a single consistent rule. Many students in Exercise 8A stumble on terms like 1/x², forgetting to write it as x⁻² first.
在求导之前，始终将根式和倒数改写为幂的形式。这样可以避免错误，并让你始终使用同一条一致的法则。许多学生在练习 8A 中遇到 1/x² 这样的项时会犯错，因为他们忘了先将其写成 x⁻²。
4. Constant Multiple Rule | 常数乘法则
When a function is multiplied by a constant, the derivative is simply that constant multiplied by the derivative of the function. In symbols:
当一个函数乘以一个常数时，导数就等于该常数乘以这个函数的导数。用符号表示为：
d/dx [c·f(x)] = c·f'(x)
For example, if f(x) = 5x³, the derivative is 5·(3x²) = 15x². The constant just "comes along for the ride" sively throughout Exercise 8A, especially when dealing with polynomials with coefficients.
例如，如果 f(x) = 5x³，其导数就是 5·(3x²) = 15x²。常数只是“随波逐流”，在微分过程中不会改变。这一法则在练习 8A 中被广泛使用，尤其是在处理带有系数的多项式时。
Be careful: the constant multiple rule applies only to multiplication by a constant, not to a product of two non-constant functions. In Exercise 8A, problems are designed to help you distinguish this from the product rule which comes later.
注意：常数乘法则仅适用于乘以一个常数的情况，而不适用于两个非常数函数的乘积。练习 8A 中的题目正是为了帮助你区分这一点和稍后会学到的积法则。
s you to differentiate any term of the form a xⁿ quickly. For instance, d/dx (7x⁴) = 28x³.
将幂法则与常数乘法则结合，你就能快速对任何 a xⁿ 形式的项进行求导。例如，d/dx (7x⁴) = 28x³。
5. Sum and Difference Rules | 和差法则
Differentiation is a linear operation, meaning the derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives.
微分是一种线性运算，这意味着和的导数等于导数的和，差的导数等于导数的差。
d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
differentiate each piece: the derivative is 6x + 2 + 0. The constant term -5 becomes zero because the slope of a horizontal line is zero.
这样你就可以对多项式逐项求导。例如，要对 y = 3x² + 2x - 5 进行求导，只需对每一部分分别求导：导数为 6x + 2 + 0。常数项 -5 的导数为零，因为水平线的斜率为零。
Exercise 8A typically provides polynomials of degree two, three and four so you can practise applying the power rule, constant multiple rule, and sum/difference rules together. Always work step by step to minimise sign errors.
练习 8A 通常会给出二次、三次和四次多项式，让你能够综合练习幂法则、常数乘法则以及和差法则。务必一步一步计算，以减少符号错误。
Once you master term-by-term differentiation, even lengthier polynomials become straightforward. The key is to keep your work organised and check each term individually.
一旦你掌握了逐项求导，即使是较长的多项式也会变得简单。关键在于保持步骤有条理，并逐一检查每一项。
6. Differentiating Standard Polynomials – Worked Examples | 标准多项式的微分——示例讲解
Let' ical problems you might encounter in Exercise 8A. Differentiate f(x) = 4x⁵ - 3x³ + 2x - 7.
我们来将法则应用到练习 8A 中可能遇到的一些典型题目上。对 f(x) = 4x⁵ - 3x³ + 2x - 7 进行求导。
Using the power rule and constant multiple rule on each term:
- Derivative of 4x⁵ is 4·5x⁴ = 20x⁴
- Derivative of -3x³ is -3·3x² = -9x²
- Derivative of 2x is 2·1x⁰ = 2
- Derivative of -7 is 0
So f'(x) = 20x⁴ - 9x² + 2.
对每一项使用幂法则和常数乘法则：
- 4x⁵ 的导数是 4·5x⁴ = 20x⁴
- -3x³ 的导数是 -3·3x² = -9x²
- 2x 的导数是 2·1x⁰ = 2
- -7 的导数是 0
因此 f'(x) = 20x⁴ - 9x² + 2。
Another example: differentiate g(x) = 2/√x + x³. Start by rewriting: g(x) = 2x-1/2 + x³. Then g'(x) = 2·(-1/2)x⁻3/2 + 3x² = -x⁻3/2 + 3x², which can also be written as -1/(x√x) + 3x².
再举一个例子：对 g(x) = 2/√x + x³ 求导。首先改写：g(x) = 2x-1/2 + x³。然后 g'(x) = 2·(-1/2)x⁻3/2 + 3x² = -x⁻3/2 + 3x²，也可以写成 -1/(x√x) + 3x²。
These types of mixed-term expressions are very common in IB Exercise 8A. Always rewrite radicals and denominators in exponent form before differentiating, and simplify the final answer as much as possible.
这种混合项表达式在 IB 练习 8A 中非常常见。求导前一定要将根式和分母改写为指数形式，并尽可能化简最终答案。
7. Finding the Equation of a Tangent Line | 求切线方程
equation of the tangent line to a curve at a given point. The tangent line has slope m = f'(a) and passes through (a, f(a)).
练习 8A 中考查的最重要的应用之一，就是利用导数求出曲线在给定点处的切线方程。切线的斜率为 m = f'(a)，并且它经过点 (a, f(a))。
n using the point-slope form: y - y₁ = m(x - x₁). Substituting x₁ = a, y₁ = f(a) and m = f'(a) gives y = f(a) + f'(a)(x - a).
切线方程可以用点斜式来写：y - y₁ = m(x - x₁)。代入 x₁ = a、y₁ = f(a) 和 m = f'(a) 就得到 y = f(a) + f'(a)(x - a)。
the tangent to y = x² - 3x + 2 at x = 1. First, f(1) = 1 - 3 + 2 = 0. The derivative is f'(x) = 2x - 3, so f'(1) = 2 - 3 = -1. The equation is y - 0 = -1(x - 1) ⇒ y = -x + 1.
例如，求曲线 y = x² - 3x + 2 在 x = 1 处的切线。首先 f(1) = 1 - 3 + 2 = 0。导数为 f'(x) = 2x - 3，所以 f'(1) = 2 - 3 = -1。切线方程为 y - 0 = -1(x - 1)，化简得 y = -x + 1。
or to determine the point of tangency given a slope. These problems link the algebraic derivative back to the geometry of the graph.
在练习 8A 中，你可能需要找出切线水平的点（m = 0），或是根据给定的斜率确定切点。这些题目将代数导数与图像的几何意义联系了起来。
8. Normal Lines to a Curve | 曲线的法线
The normal line to a curve at a point is perpendicular to the tangent at that point. If the tangent has slope m, the normal has slope -1/m (provided m ≠ 0). This concept appears frequently alongside tangent problems in IB exercises.
曲线在某一点的法线垂直于该点处的切线。如果切线的斜率为 m，那么法线的斜率就是 -1/m（前提是 m ≠ 0）。在 IB 练习中，法线问题经常和切线问题一起出现。
To find the equation of the normal, first compute the derivative to find m = f'(a). Then the normal slope is -1/f'(a). Use the same point (a, f(a)) and write the equation in point-slope form.
要求出法线方程，首先计算导数以求出 m = f'(a)。然后法线的斜率就是 -1/f'(a)。利用相同的点 (a, f(a)) 并以点斜式写出方程。
Example: For y = x² at x = 2, f' oint is (2, 4). The normal equation is y - 4 = -1/4 (x - 2). Simplify to y = -1/4 x + 4.5.
示例：对于 y = x² 在 x = 2 处，f'(x) = 2x，因此 m = 4。法线斜率为 -1/4，点为 (2, 4)。法线方程为 y - 4 = -1/4 (x - 2)，化简得 y = -1/4 x + 4.5。
Students often confuse the normal with the tangent or forget to take the negative reciprocal. Exercise 8A includes problems that train you to switch quickly between the two concepts.
学生们常常将法线与切线混淆，或是忘记取负倒数。练习 8A 包含了训练你在两个概念之间快速切换的题目。
9. The Second Derivative | 二阶导数
The second derivative, denoted f''(x) or d²y/dx², is simply the derivative of the first derivative. It measures the rate of change of the slope, giving information about the concavity of the graph and acceleration in motion problems.
二阶导数，记作 f''(x) 或 d²y/dx²，就是一阶导数的导数。它衡量斜率的变化率，提供了关于图形凹凸性以及运动问题中加速度的信息。
differentiate the function twice. For f(x) = 2x³ - 5x² + 3x - 1, the first derivative is f'(x) = 6x² - 10x + 3. The second derivative is f''(x) = 12x - 10.
在练习 8A 中求多项式的二阶导数，需要对函数进行两次求导。对于 f(x) = 2x³ - 5x² + 3x - 1，一阶导数为 f'(x) = 6x² - 10x + 3，二阶导数为 f''(x) = 12x - 10。
Second derivatives help determine whether a stationary point is a maximum or minimum: if f''(a) > 0, the curve is concave up and the point is a minimum; if f''(a) < 0, it is a maximum. This is called the second derivative test.
二阶导数有助于判断驻点是极大值还是极小值：如果 f''(a) > 0，曲线是下凸的，该点为极小值点；如果 f''(a) < 0，则为极大值点。这被称为二阶导数判别法。
Exercise 8A might introduce you to simple second derivative calculations and basic interpretation of concavity. Even if not fully explored, it sets the stage for optimisation problems later in the course.
练习 8A 可能会向你介绍简单的二阶导数计算以及对凹凸性的基本解释。即使没有展开讨论，也会为后续课程中的最优化问题打下基础。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Many learners slip up in Exercise 8A by forgetting to apply the power rule correctly to negative or fractional exponents. Always rewrite terms before differentiating and double-check the exponent after subtracting 1.
许多学习者在练习 8A 中会因为忘记对负指数或分数指数正确应用幂法则而出错。务必在求导前重写各项，并在减去 1 后仔细检查指数。
Another common error is mishandling constant terms. A constant like 5 or -π always differentiates to 0. Avoid the temptation to carry the constant through as if it were a coefficient of x.
另一个常见错误是处理常数项不当。常数如 5 或 -π 的导数始终为零。不要试图像处理 x 的系数那样将常数保留下来。
Sign errors when differentiating negative coefficients also plague students. For example, the derivative of -4x² is -8x, not 8x. Take your time with each term and maintain a clear notation.
在对负系数求导时出现符号错误也困扰着学生。例如，-4x² 的导数是 -8x，而不是 8x。要对每一项从容计算，并保持清晰的书写。
te x = a into the original function to get y₁, and into the derivative to get the slope m.
在求切线时，一些学生会误将导数用作 y 值，或者将 x 代入错误的表达式。记住：将 x = a 代入原函数以得到 y₁，代入导数以得到斜率 m。
Exercise 8A is designed to help you internalise these fundamentals through repetition. Review your mistakes carefully – they often highlight the difference between a superficial understanding and true mastery.
练习 8A 旨在通过重复练习帮助你内化这些基础知识。仔细回顾你的错误——它们通常能揭示肤浅理解与真正掌握之间的差距。
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