📚 Exercise 16A: Mastering Derivative Rules | 练习16A：掌握导数法则
Exercise 16A in your IB Mathematics course book is designed to build solid foundations in differentiation. It focuses on the core rules that will allow you to differentiate a wide variety of functions without having to revert to first principles each time. By working through this exercise carefully, you will gain fluency in the power rule, product rule, quotient rule, chain rule and the derivatives of trigonometric functions.
IB 数学教材中的练习 16A 旨在帮助你牢固掌握微分基础。它聚焦于核心求导法则，让你能够快速对多种函数求导，而无需每次都从第一原理出发。认真完成这部分练习，你将熟练运用幂法则、乘积法则、商法则、链式法则以及三角函数的导数。
1. Understanding the Power Rule | 理解幂法则
the most fundamental tool in differentiation. For any real number n, if f(x) = xⁿ, then f' integers, negative integers and even fractional exponents, making it incredibly versatile.
幂法则是微分中最基本的工具。对于任意实数 n，如果 f(x) = xⁿ，那么 f'(x) = n xⁿ⁻¹。这一法则适用于正整数、负整数甚至分数指数，用途极为广泛。
For example, the derivative of x⁵ is 5x⁴. For x⁻³, you obtain -3x⁻⁴. When dealing with roots like √x, rewrite as x½ to get (1/2)x⁻½, or 1/(2√x).
例如，x⁵ 的导数是 5x⁴。对 x⁻³ 求导得到 -3x⁻⁴。处理根式如 √x 时，将其改写为 x½，就能得到 (1/2)x⁻½，即 1/(2√x)。
2. The Constant Multiple Rule | 常数倍法则
tiation operation. If c is a constant and f(x) is differentiable, then d/dx [c·f(x)] = c·f'(x).
常数倍法则允许你在微分运算中将常数系数提取出来。如果 c 是常数且 f(x) 可导，那么 d/dx [c·f(x)] = c·f'(x)。
s rule frequently combines with the power rule and saves you from expanding unnecessarily.
例如，7x⁴ 的导数就是 7·(4x³) = 28x³。这一法则常与幂法则联合使用，能避免不必要的展开。
3. Sum and Difference Rules | 和差法则
The sum rule states that the derivative of a sum is the sum of the derivatives: (f + g)' = f' + g' lly with a minus sign. Together, they allow you to differentiate polynomials term by term.
和法则表明，和的导数等于导数的和：(f + g)' = f' + g'。差法则同理，只是改用减号。两者结合，就可以对多项式逐项求导。
Consider h(x) = 3x² - 5x + 2. Using the power rule and the sum/difference rules, h'(x) = 6x - 5. The constant 2 disappears because the derivative of any constant is zero.
考虑 h(x) = 3x² - 5x + 2。运用幂法则以及和差法则，得到 h'(x) = 6x - 5。常数项 2 的导数为零。
4. Differentiating Polynomials | 多项式求导
Polynomial differentiation is a direct application of the rules seen so far. A general polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ has derivative n aₙxⁿ⁻¹ + (n-1)aₙ₋₁xⁿ⁻² + ... + a₁. There is no need to write first principles every time.
多项式求导是前面几条法则的直接应用。一般多项式 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ 的导数为 n aₙxⁿ⁻¹ + (n-1)aₙ₋₁xⁿ⁻² + ... + a₁。无需每次都从第一原理推导。
In Exercise 16A, you will meet polynomials of degree 2, 3 and higher. Always double-check the new exponent after differentiation; it is easy to forget to subtract 1.
在练习 16A 中，你会遇到二次、三次及更高次的多项式。务必仔细检查求导后的新指数，忘记减 1 是最常见的粗心错误之一。
5. The Product Rule | 乘积法则
ntiate a product of two functions, the product rule is essential. If y = u(x)·v(x), then y' = u'v + uv' oes not matter for addition, but you must keep the original functions intact in each term.
当需要对两个函数的乘积求导时，乘积法则必不可少。若 y = u(x)·v(x)，则 y' = u'v + uv'。加法项的顺序无关紧要，但每一项中必须保留原函数。
A typical exercise from 16A might include f(x) = x² sin x. Here u = x², v = sin x, so f'(x) = 2x sin x + x² cos x. Recognising when to apply the product rule takes practice.
练习 16A 中的典型题目可能包含 f(x) = x² sin x。这里 u = x²，v = sin x，于是 f'(x) = 2x sin x + x² cos x。何时该用乘积法则，需要大量练习来判断。
6. The Quotient Rule | 商法则
he quotient rule handles division of two functions. For y = u(x)/v(x), the derivative is y' = (u'v - uv' 0. The numerator must respect the subtraction order, as swapping u and v changes the sign.
商法则处理两个函数相除的情况。对于 y = u(x)/v(x)，导数为 y' = (u'v - uv')/v²，前提是 v(x) ≠ 0。分子必须遵循减法顺序，交换 u 和 v 会改变符号。
For g(x) = (3x + 1)/(x² - 4), set u = 3x+1, v = x²-4, then g'(x) = (3(x²-4) - (3x+1)(2x)) / (x²-4)². Simplify carefully to avoid algebraic slips.
对于 g(x) = (3x + 1)/(x² - 4)，令 u = 3x+1，v = x²-4，得 g'(x) = (3(x²-4) - (3x+1)(2x)) / (x²-4)²。化简时要格外细心，避免代数失误。
7. Derivatives of Trigonometric Functions | 三角函数的导数
IB Mathematics expects you to know the derivatives of sin, cos and tan by heart. The basic results are:
| f(x) | f'(x) |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
IB 数学要求学生牢记 sin、cos 和 tan 的导数。基本结果如上表所示。这些结果可以在本章后续部分通过商法则进一步推导出来。
Remember that angle arguments are always in radians when differentiating. A function like sin(2x) will involve the chain rule, which appears later in the exercise.
记住，求导时角度始终要以弧度为单位。像 sin(2x) 这样的函数会涉及链式法则，这会在练习后续部分出现。
8. The Chain Rule Introduction | 链式法则初探
The chain rule is used for composite functions y = f(g(x)). It states that dy/dx = f'(g(x))·g' is rule multiplies the rate of change of the outer function by that of the inner function.
链式法则用于复合函数 y = f(g(x))。它表明 dy/dx = f'(g(x))·g'(x)，或用莱布尼茨记号写为 dy/dx = (dy/du)·(du/dx)，其中 u = g(x)。这一法则将外部函数的变化率与内部函数的变化率相乘。
For h(x) = (2x + 3)⁴, choose u = 2x+3, then h'(x) = 4u³·2 = 8(2x+3)³. Spotting the inner function is the first and most crucial step.
对于 h(x) = (2x + 3)⁴，选取 u = 2x+3，则 h'(x) = 4u³·2 = 8(2x+3)³。识别出内部函数是关键的第一步。
9. Common Mistakes in Exercise 16A | 练习16A中的常见错误
One frequent error is forgetting to apply the chain rule when the argument is not simply x. For instance, differentiating cos(5x) as -sin(5x) without multiplying by 5 is incorrect.
一个常见错误是当自变量不仅仅是 x 时，忘记使用链式法则。例如，将 cos(5x) 的导数写作 -sin(5x) 而未乘以 5 就是错误的。
Another slip is miswriting the quotient rule numerator as uv' - u'v instead of u'v - uv'. Also, students sometimes cancel terms prematurely before expanding, which can lead to sign errors.
另一个失误是将商法则的分子错写成 uv' - u'v，而不是正确的 u'v - uv'。此外，有时学生在展开之前就过早约分，导致符号错误。
Power-rule misapplication with negative exponents is also common: the derivative of x⁻² is -2x⁻³, but many miscalculate the new exponent as -1 or -3 with a wrong sign.
负指数幂法则的应用错误也很普遍：x⁻² 的导数是 -2x⁻³，但许多人会把新指数误算为 -1，或者符号算错。
10. Step-by-Step Worked Examples | 分步解题示例
Let' term. The derivative of 3x⁴ is 12x³; of -2x² is -4x; and the constant 5 becomes 0. Thus f'(x) = 12x³ - 4x.
让我们演练一道练习 16A 中的典型题：对 f(x) = 3x⁴ - 2x² + 5 求导。第一步：逐项应用幂法则。3x⁴ 的导数是 12x³；-2x² 的导数是 -4x；常数 5 的导数为 0。因此 f'(x) = 12x³ - 4x。
Consider g(x) = x² cos x. Here we use the product rule with u = x², v = cos x. Then u' = 2x, v' = -sin x. Substituting: g'(x) = 2x cos x + x² (-sin x) = 2x cos x - x² sin x.
考虑 g(x) = x² cos x。这里需用乘积法则，取 u = x²，v = cos x。则 u' = 2x，v' = -sin x。代入得：g'(x) = 2x cos x + x² (-sin x) = 2x cos x - x² sin x。
For a quotient, try h(x) = (x²+1)/(2x-3). Set u = x²+1, v = 2x-3, giving u' = 2x, v' = 2. Then h' x(2x-3) - (x²+1)·2) / (2x-3)² = (4x² - 6x - 2x² - 2) / (2x-3)² = (2x² - 6x - 2) / (2x-3)².
对于商，尝试 h(x) = (x²+1)/(2x-3)。令 u = x²+1，v = 2x-3，得到 u' = 2x，v' = 2。于是 h' 2x(2x-3) - (x²+1)·2) / (2x-3)² = (4x² - 6x - 2x² - 2) / (2x-3)² = (2x² - 6x - 2) / (2x-3)²。
11. Tips for Checking Your Answers | 检查答案的技巧
rivative expression and the function, and compare the estimated slope from a small difference in x. This numeric check can quickly flag a mistake.
做完一道题后，代入一个简单数值，比如 x = 1，分别计算导数的值和通过微小的 x 增量估算的斜率，两者应接近。这种数值检验能快速发现错误。
Graphical verification is also powerful: on your graphing calculator, plot the original function and the derivative you found. The derivative should be positive where the function is increasing, and zero at turning points.
图像验证也很有效：在图形计算器上画出原函数和你求出的导函数。导函数应在原函数上升时为正，在驻点处为零。
If the exercise provides final answers at the back of the book, do not look before a genuine attempt. After comparing, re-do any incorrect problems from scratch without peeking at the solution.
如果练习在书末附有答案，不要过早查看。先真正尝试解答，核对后在完全不看解答的情况下重新演算每一道错题。
12. Relating to IB Exam Questions | 与IB考题的联系
Exercise 16A mirrors the style of Paper 1 and Paper 2 questions that require straightforward differentiation. Examiners often award marks for the correct use of rules, even if the final simplified form is partly incomplete.
练习 16A 的题风与试卷一和试卷二中对直接求导的考题相似。阅卷人通常看重正确运用求导规则，即使最终化简形式不完全到位也能得到方法分。
In Section A of the exam, you might be asked to differentiate a product or quotient as the first step in a larger problem involving tangents, normals or optimisation. Mastering the rules here greatly reduces time pressure later.
在考试 A 部分，你可能需要在求切线、法线或优化问题中先求出某个乘积或商的导数。在此扎实地掌握这些法则，将极大缓解后续的时间压力。
Additionally, familiarity with the chain rule and trigonometric derivatives gives you the confidence to tackle integration by inspection later in the course, reinforcing the connection between derivative and antiderivative.
此外，熟悉链式法则和三角函数的导数能让你在后续课程中自信地应对观察积分，进一步巩固导数与反导数之间的关系。
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