📚 Exercise 6E: Chain Rule for Derivatives | 练习6E：导数的链式法则
In IB Mathematics, differentiating simple power functions is only the beginning. When functions are built from inside another, such as (3x² - 2)⁵ or sin(5x), the ordinary power rule will not work on its own. Exercise 6E is specifically designed to build fluency with the chain rule - the technique that unlocks derivatives of composite functions. Understanding this exercise thoroughly will give you confidence to handle related rates, tangents, and advanced integration later in the course.
在 IB 数学中，对简单幂函数求导只是起点。当函数由一个嵌套在另一个内部构成时，比如 (3x² - 2)⁵ 或 sin(5x)，单纯的幂法则便不再奏效。练习 6E 旨在训练链式法则的熟练运用，正是这套技巧能够解锁复合函数的导数。透彻掌握这部分练习，将会让你在后续的变化率、切线以及更深入的积分内容中充满自信。
1. Recognising Composite Functions | 识别复合函数
x + 1 is placed inside the outer function u⁴. Without a clear recognition of which part is ‘inside’ and which is ‘outside’, the chain rule cannot be applied correctly.
复合函数是指一个完整的表达式被放在另一个运算内部。例如，y = (2x + 1)⁴ 并不只是 x 的幂；内层函数 u = 2x + 1 被放在了外层函数 u⁴ 的内部。如果不能清晰地识别哪部分是‘内层’、哪部分是‘外层’，链式法则就无法正确使用。
Consider y = √cos x re root, and the inner function is cos x. In Leibniz notation this might be written as y = √u ction in terms of an intermediate variable is the first step in many Exercise 6E problems.
例如 y = √cos x，这里外层函数是平方根，内层函数是 cos x。用莱布尼兹记号可以记作 y = √u，其中 u = cos x。将函数重新用中间变量表达，是解决练习 6E 中许多问题的第一步。
2. The Chain Rule: Two Standard Forms | 链式法则：两种标准形式
in rule can be memorised in two equivalent ways. In prime notation, if y = f(g(x)), then y' = f'(g(x)) · g' x. The Leibniz form makes it especially clear that we are multiplying the rates of change.
链式法则可以用两种等价的形式记忆。用撇号记法，若 y = f(g(x))，则 y' = f'(g(x)) · g'(x)。用莱布尼兹记号，法则是 dy/dx = dy/du × du/dx，这里 y 是 u 的函数，u 是 x 的函数。莱布尼兹形式特别清晰地表明，我们是在将变化率相乘。
e y with respect to u, multiply by du/dx, and then replace u. This structured approach reduces mistakes in algebraic manipulation.
在解答练习 6E 时，很多学生倾向于写出替换 u = 内层函数，求出 y 关于 u 的导数，乘以 du/dx，再回代 u。这种结构化的方法可以减少代数操作中的错误。
3. Applying the Chain Rule to Polynomial-Like Expressions | 链式法则在类多项式表达式中的应用
/du = 6u⁵ and du/dx = 15x². Multiplying gives dy/dx = 6(5x³ - 7)⁵ × 15x² = 90x²(5x³ - 7)⁵.
练习 6E 中最常见的题型是要求对类似 y = (5x³ - 7)⁶ 的函数求导。首先，确定外层函数为 u⁶，内层函数为 u = 5x³ - 7。然后 dy/du = 6u⁵，du/dx = 15x²。相乘得到 dy/dx = 6(5x³ - 7)⁵ × 15x² = 90x²(5x³ - 7)⁵。
Remember to always leave the inner function unchanged when writing the derivative of the outer function. A common slip is to square or alter the inside prematurely - only after multiplying by du/dx should you consider further simplification.
请记住，在写出外层函数的导数时要保持内层函数不变。一个常见的差错就是过早地对内部进行平方或修改——只有乘以 du/dx 之后，才应该进一步化简。
4. Differentiating Trigonometric Composite Functions | 三角复合函数的微分
sites. For y = sin(4x - 1), the outer derivative of sin u is cos u, and du/dx = 4. Hence y' = cos(4x - 1) × 4 = 4 cos(4x - 1). With y = tan(3x²), the derivative is sec²(3x²) × 6x.
三角函数经常出现在复合结构中。对于 y = sin(4x - 1)，外层 sin u 的导数是 cos u，而 du/dx = 4。因此 y' = cos(4x - 1) × 4 = 4 cos(4x - 1)。对于 y = tan(3x²)，导数则是 sec²(3x²) × 6x。
u² gives 2 sin(2x), then multiplied by the derivative of sin(2x) which is 2 cos(2x), so y' = 3×2 sin(2x)×2 cos(2x) = 12 sin(2x) cos(2x). Recognising the multiple layering is vital.
练习 6E 还可能包含稍微复杂的情形，例如 y = 3 sin²(2x)。这里需要使用两次链式法则：外层幂 u² 给出 2 sin(2x)，再乘以 sin(2x) 的导数 2 cos(2x)，因此 y' = 3×2 sin(2x)×2 cos(2x) = 12 sin(2x) cos(2x)。识别出多层嵌套至关重要。
5. Chain Rule with Exponential and Logarithmic Functions | 涉及指数与对数函数的链式法则
tions often keep their basic form but pick up an extra factor from the exponent. For y = e2x+1, the derivative is e2x+1 × 2 = 2e2x+1. For y = 53x, rewrite as e3x ln 5, then y' = 3 ln 5 × 53x. In Exercise 6E you will see many variations of ekx and abx.
指数函数求导通常保持基本形式，但从指数处额外获得一个因子。对于 y = e2x+1，导数为 e2x+1 × 2 = 2e2x+1。对于 y = 53x，可改写为 e3x ln 5，则 y' = 3 ln 5 × 53x。在练习 6E 中你会见到多种 ekx 和 abx 的变式。
1, then dy/dx = (1/(4x²+1)) × 8x = 8x/(4x²+1). The numerator always becomes exactly du/dx.
处理自然对数时，记住链式法则与 ln u 的导数 1/u 搭配使用。所以对于 y = ln(4x² + 1)，令 u = 4x² + 1，则 dy/dx = (1/(4x²+1)) × 8x = 8x/(4x²+1)。分子总是恰好为 du/dx。
6. Applying the Chain Rule Multiple Times | 多次使用链式法则
Functions like y = sin(ecos x) require the chain rule to be applied again and again. The outer function is sin, its argument is ecos x. The derivative of sin is cos(ecos x), then multiplied by the derivative of ecos x, which is ecos x × (-sin x). Altogether, y' = cos(ecos x) × ecos x × (-sin x).
像 y = sin(ecos x) 这样的函数需要反复应用链式法则。最外层函数是 sin，其自变量为 ecos x。sin 的导数是 cos(ecos x)，再乘以 ecos x 的导数，即 ecos x × (-sin x)。合在一起，y' = cos(ecos x) × ecos x × (-sin x)。
Exercise 6E frequently combines polynomial, trigonometric, exponential and logarithmic components within one question. A systematic approach—labelling u₁, u₂, etc.—can prevent confusion when the chain is deep. Always work from the outermost layer inwards.
练习 6E 经常将多项式、三角、指数和对数成分组合在一道题中。当嵌套层次很深时，系统地标记 u₁ 和 u₂ 等，可以防止混乱。一定要从最外层向里逐层求导。
7. Using the Chain Rule to Find Tangents and Normals | 用链式法则求切线和法线
ain rule, you can immediately write the equation of a tangent line. For example, given y = √x² + 9, find the tangent at x = 4. First, dy/dx = (1/(2√x²+9)) × 2x = x / √x²+9. At x = 4, dy/dx = 4/5, and y = 5, so the tangent is y - 5 = (4/5)(x - 4).
通过链式法则求出导数后，你可以立即写出切线方程。例如，给定 y = √x² + 9，求 x = 4 处的切线。首先，dy/dx = (1/(2√x²+9)) × 2x = x / √x²+9。在 x = 4 处，dy/dx = 4/5，y = 5，因此切线为 y - 5 = (4/5)(x - 4)。
For a normal line, remember the gradient is the negative reciprocal. If the question involves composite trigonometric functions, evaluate the derivative carefully, ensuring your calculator is in radian mode as expected for calculus. Exercise 6E often includes coordinate geometry follow-ups to make the derivatives practical.
对于法线，要记住斜率是负倒数。如果题目涉及复合三角函数，计算导数时要格外仔细，确保计算器处于微积分所需的弧度模式。练习 6E 经常会搭配坐标几何的后续问题，使求导更具实际意义。
8. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One frequent error is forgetting to multiply by du/dx entirely. A student might write the derivative of (3x+2)⁵ as 5(3x+2)⁴ and stop there. Always check whether an inner derivative factor is missing—if the inner function is not simply x, there is almost certainly an extra factor.
一个常见的错误是完全忘记乘以 du/dx。学生可能会将 (3x+2)⁵ 的导数写成 5(3x+2)⁴ 就结束了。务必检查是否遗漏了内层导数的因子——如果内层函数不是单纯的 x，几乎肯定存在一个额外的因子。
ct and quotient rules with the chain rule. For y = x²(2x+1)³, you need the product rule: y' + x²·3(2x+1)²·2. Always identify the overall structure before diving into differentiation.
另一个陷阱是对乘积错误地使用链式法则。在练习 6E 中，有些题目会将积法则、商法则与链式法则混合。对于 y = x²(2x+1)³，你需要使用积法则：y' = 2x(2x+1)³ + x²·3(2x+1)²·2。在开始求导之前，一定要先识别整体结构。
9. Moving to Abstract and Symbolic Problems | 向抽象和符号化问题进阶
essions like y = f(g(x)) given only graphical or numerical information. For instance, if f'(3) = 2, g(2) = 3 and g'(2) = 4, then the derivative of f(g(x)) at x = 2 is f'(g(2))·g'(2) = f'(3)·4 = 2·4 = 8. Such questions test conceptual understanding.
练习 6E 通常会过渡到只需图形或数值信息便能对 y = f(g(x)) 这类表达式求导的问题。例如，若已知 f'(3) = 2，g(2) = 3 且 g'(2) = 4，则 f(g(x)) 在 x = 2 处的导数为 f'(g(2))·g'(2) = f'(3)·4 = 2·4 = 8。这类题目考查的是概念性理解。
Table-based problems also appear: a table of values for f, g, f' and g' is given, and you must combine the chain rule with other rules. The key is to substitute carefully and not to confuse which derivative is being evaluated at which point. Write down the chain rule formula before inserting numbers.
基于表格的问题也会出现：给出关于 f、g、f' 和 g' 的数值表，你必须将链式法则与其他法则结合使用。关键是要仔细代入，不要混淆哪个导数在哪个点取值。在代入数字之前，先写下链式法则的公式。
10. Summary and Practice Mindset | 总结与练习心态
Mastering Exercise 6E means internalising the habit of looking inside every non-basic function for an inner core. The chain rule is not an isolated topic; it is the glue that connects all differentiation techniques. Once you can apply it automatically, related rates, implicit differentiation and integration by substitution become far more accessible.
掌握练习 6E 意味着内化一种习惯：面对每一个非常规函数，都要去寻找它内部的‘芯’。链式法则并非孤立的话题，它是连接所有微分技巧的粘合剂。一旦你能不加思索地运用它，变化率、隐函数微分法和换元积分法都将变得容易许多。
Approach the exercise by first working through straightforward polynomial composites, then adding trigonometric, exponential, and logarithmic layers one at a time. Check your solutions by mentally verifying that the inner derivative has been included, and remember that the IB examiner looks for clear algebraic steps. With consistent practice, the chain rule will become second nature.
练习时，先从简单的多项式复合入手，然后一次只叠加一层三角函数、指数或对数。检查答案时，在脑中验算是否包含了内层导数，并记住 IB 考官看重清晰的代数步骤。通过持续练习，链式法则将成为你的第二本能。
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