📚 Exercise 10D: Applications of Differentiation – Tangents and Normals | 练习10D：微分应用 – 切线与法线
In Exercise 10D, students apply the rules of differentiation to find equations of tangents and normals to curves. This topic builds directly on the concept that the derivative represents the gradient of a curve at a point. By mastering this exercise, you will learn to calculate instantaneous rates of change and use them to construct linear equations that describe the local behaviour of a function. The skills practised here are essential for IB Mathematics Analysis & Approaches and are frequently assessed in Paper 1 and Paper 2.
在练习10D中，学生将运用微分法则求解曲线的切线与法线方程。该主题直接建立在导数表示曲线在某点处斜率的概念之上。通过掌握这个练习，你将学会计算瞬时变化率，并利用这些变化率构建描述函数局部行为的线性方程。这里训练的解题技巧对IB数学分析与方法至关重要，经常出现在试卷一和试卷二中。
1. The Derivative as a Gradient Function | 导数作为斜率函数
The derivative of a function f(x), written f' the gradient of the tangent to the curve y = f(x) at any point x. In geometrical terms, f' This interpretation is the foundation for all tangent and normal problems in Exercise 10D.
函数 f(x) 的导数，记作 f'(x) 或 dy/dx，给出了曲线 y = f(x) 在任意点 x 处切线的斜率。从几何角度来看，f'(a) 就是刚好在 x = a 处与曲线相切的直线的斜率。这种几何解释是练习10D中所有切线与法线问题的基础。
two points get infinitely close. This is exactly the limit definition of the derivative: f'(x) = limh→0 [f(x+h) – f(x)] / h. In IB exams, however, you will mainly use the differentiation rules to find derivatives quickly.
当我们从割线（穿过两点）过渡到切线时，我们取两点无限接近的极限。这正是导数的极限定义：f'(x) = limh→0 [f(x+h) – f(x)] / h。不过在IB考试中，你主要会用求导法则来快速得到导数。
2. Review of Basic Differentiation Rules | 回顾基本微分法则
t with the standard differentiation rules. The power rule states that if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. The constant multiple rule tells us that the derivative of c·f(x) is c·f'(x). The sum rule allows us to differentiate term by term: (f + g)' = f' + g'.
在处理切线和法线问题之前，你必须熟练运用标准微分法则。幂法则指出，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。常数倍法则告诉我们，c·f(x) 的导数是 c·f'(x)。和差法则允许我们逐项微分：(f + g)' = f' + g'。
alone allow you to differentiate any polynomial. For example, the derivative of f(x) = 4x³ – 2x² + 7x – 5 is f'(x) = 12x² – ccurately, as they are the building blocks for the more complex functions in Exercise 10D.
仅凭这三条法则，你就可以对任意多项式求导。例如，f(x) = 4x³ – 2x² + 7x – 5 的导数是 f'(x) = 12x² – 4x + 7。请确保能快速准确地完成这些运算，因为它们是练习10D中更复杂函数的基石。
d/dx (xⁿ) = n xⁿ⁻¹ | d/dx (c) = 0 | d/dx (c·f) = c·f' | (f ± g)' = f' ± g'
3. The Chain Rule | 链式法则
= f(u) and u = g(x), then dy/dx = dy/du × du/dx. In function notation, the derivative is f'(g(x)) × g'(x). This rule is indispensable for handling expressions like (2x + 1)⁵ or √3x² – x.
链式法则用于求形如 f(g(x)) 的复合函数的导数。用莱布尼茨记号表示，若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。用函数记号表示，导数为 f'(g(x)) × g'(x)。处理诸如 (2x + 1)⁵ 或 √3x² – x 这类表达式时，链式法则是必不可少的。
A typical IB problem might ask: Differentiate y = (5x – 2)⁴. Setting u = 5x – 2 gives dy/du = 4u³ and du/dx = 5, so dy/dx = 20(5x – ific point for Exercise 10D, you can substitute the x-value directly into this derivative.
IB真题中常出现类似问题：对 y = (5x – 2)⁴ 求导。令 u = 5x – 2，则 dy/du = 4u³，du/dx = 5，因此 dy/dx = 20(5x – 2)³。当你在练习10D中需要某点处的斜率时，只需将 x 值直接代入这个导数即可。
4. The Product Rule | 乘积法则
he product of two simpler functions, say y = u(x) v(x), we use the product rule: dy/dx = u' v + u v' he product is not the product of the derivatives. Many students lose marks by writing (uv)' = u' v', which is incorrect.
当函数是两个较简单函数的乘积时，例如 y = u(x) v(x)，就要使用乘积法则：dy/dx = u' v + u v'。注意顺序，并记住乘积的导数不等于导数的乘积，这一点很重要。许多学生因错误地写成 (uv)' = u' v' 而丢分。
For instance, to differentiate f(x) = x² sin x, let u = x², v = sin x. Then u' = 2x, v' = cos x. Applying the product rule gives f' stions, such a derivative is evaluated at a given x-coordinate to find the required slope.
例如，对 f(x) = x² sin x 求导时，令 u = x², v = sin x，则 u' = 2x, v' = cos x。应用乘积法则得到 f'(x) = 2x sin x + x² cos x。在切线与法线问题中，这样的导数会在给定的 x 坐标处被求值，以得到所需的斜率。
5. The Quotient Rule | 商法则
function is expressed as a quotient y = u(x) / v(x), the derivative is given by dy/dx = (u' v – u v' he subtraction order; reversing it will change the sign and lead to an incorrect gradient.
若函数以商的形式 y = u(x) / v(x) 给出，导数为 dy/dx = (u' v – u v') / v²。一个有用的记忆口诀是“底乘顶导减顶乘底导，除以底的平方”。要特别注意减法顺序；一旦颠倒就会改变符号，导致斜率错误。
For example, the derivative of f(x) = (x + 1) / (x – 2) uses u = x + 1, v = x – 2. Then u' = 1, v' = 1, and f'(x) = (1·(x – 2) – (x + 1)·1) / (x – 2)² = –3 / (x – ays simplify the numerator before substituting a point, as this reduces arithmetic errors.
例如，f(x) = (x + 1) / (x – 2) 的导数中，取 u = x + 1, v = x – 2。那么 u' = 1, v' = 1，f'(x) = (1·(x – 2) – (x + 1)·1) / (x – 2)² = –3 / (x – 2)²。在代入某点前一定要化简分子，这样可以减少算术错误。
6. Equation of a Tangent Line | 切线方程
: (1) Calculate the y-coordinate by evaluating f(a). (2) Differentiate f(x) and evaluate f'(a) to obtain the gradient m. (3) Use the point–gradient form y – f(a) = m (x – a). Simplify the equation to the required form, usually ax + by + c = 0 or y = mx + c.
要求曲线 y = f(x) 在 x = a 处切线的方程，请按以下三个步骤进行：(1) 计算 f(a) 得到 y 坐标。(2) 对 f(x) 求导并求 f'(a) 的值，得到斜率 m。(3) 使用点斜式 y – f(a) = m (x – a)。将方程化简为所需形式，通常为 ax + by + c = 0 或 y = mx + c。
The point–gradient approach works for any differentiable function. It is essential to substitute the x-value only after differentiating; a common IB exam trap is to substitute the number first and then differentiate a constant, producing zero. Always remember: differentiate first, then substitute.
点斜式适用于任何可微函数。关键在于求导之后才能代入 x 值；IB考试中一个常见的陷阱是先代入数值，然后对一个常数求导，结果得到零。请务必牢记：先求导，再代值。
7. Equation of a Normal Line | 法线方程
tangent at that point. If the tangent has gradient m (with m ≠ 0), the normal has gradient –1/m. The steps are identical to those for the tangent, except that after finding m = f'(a), you use mN = –1/m in the point–gradient formula. The final equation is y – f(a) = (–1/m) (x – a).
曲线在某点处的法线垂直于该点处的切线。如果切线斜率为 m（m ≠ 0），则法线斜率为 –1/m。解题步骤与切线完全相同，只是在求得 m = f'(a) 之后，在点斜式中使用 mN = –1/m。最终方程为 y – f(a) = (–1/m) (x – a)。
ith equation y = f(a). These cases are less frequent in Exercise 10D but are still tested.
留意特殊情况：如果切线是水平的（m = 0），则法线是垂直的，其方程简单地为 x = a。反之，如果切线是垂直的（导数未定义），则法线是水平的，方程为 y = f(a)。这些情况在练习10D中出现较少，但仍然会被考查。
8. Worked Examples | 实例解析
Example 1: Find the equation of the tangent to the curve y = x³ – 3x at the point where x = 2.
Step 1: y-coordinate: f(2) = 8 – 6 = 2.
Step 2: f'(x) = 3x² – 3, so f'(2) = 12 – 3 = 9.
Step 3: y – 2 = 9(x – 2) → y = 9x – 16.
例1：求曲线 y = x³ – 3x 在 x = 2 处的切线方程。
步骤1：y坐标：f(2) = 8 – 6 = 2。
步骤2：f'(x) = 3x² – 3，故 f'(2) = 12 – 3 = 9。
步骤3：y – 2 = 9(x – 2) → y = 9x – 16。
Example 2: Determine the normal line to y = √x at x = 4.
y-coordinate: f(4) = 2. Derivative: f'(x) = 1/(2√x), so f'(4) = 1/4. Tangent gradient m = 1/4, thus normal gradient = –4. Normal equation: y – 2 = –4(x – 4) → y = –4x + 18.
例2：求曲线 y = √x 在 x = 4 处的法线方程。
y坐标：f(4) = 2。导数：f'(x) = 1/(2√x)，故 f'(4) = 1/4。切线斜率 m = 1/4，因此法线斜率 = –4。法线方程：y – 2 = –4(x – 4) → y = –4x + 18。
9. Common Mistakes and Pitfalls | 常见错误与陷阱
One of the most frequent errors is substituting the x-value before differentiating, which leads to a gradient of zero and an invalid tangent equation. Another common mistake is mishandling the sign when using the quotient rule or when taking the negative reciprocal for the normal. Some students forget to apply the chain rule correctly for composite functions, resulting in an incomplete derivative.
最常见的错误之一是在求导前就代入 x 值，这会导致斜率为零，切线方程无效。另一个常见错误是使用商法则时弄错符号，或在求法线斜率时忘记取负倒数。一些学生忘记对复合函数正确使用链式法则，导致导数不完整。
Check your arithmetic carefully when calculating f(a) and f'(a). Even a minor slip in evaluating a power or a product can propagate and produce a completely wrong line equation. In exam conditions, it is wise to verify your tangent equation by checking that the original point satisfies it.
在计算 f(a) 和 f'(a) 时要仔细检查算术。哪怕幂运算或乘积求值时的一个小错误，也可能导致最终的直线方程完全错误。在考试中，最好通过验证原点是否满足切线方程来检查结果的合理性。
10. Tips for Success in Exercise 10D | 练习10D成功技巧
Practice differentiating a wide variety of functions – polynomials, rational functions, radicals written as fractional powers, and trigonometric combinations. For tangent/normal problems, always label the point clearly and write down the gradient explicitly before forming the equation. Create a checklist: differentiate, substitute point, compute gradient, apply point–gradient form, simplify.
大量练习不同类型函数的求导——多项式、有理函数、写成分数指数幂的根式，以及三角函数的组合。对于切线与法线问题，务必清楚标明点坐标，并在建立方程前明确写出斜率。创建一个检查清单：求导、代入点、计算斜率、应用点斜式、化简。
When dealing with normals, draw a quick sketch of the tangent and the perpendicular relationship. This visual check often prevents the mistake of forgetting to flip the gradient. Finally, work through past IB paper questions that combine differentiation with tangents and normals, as they frequently appear in Section A and Section B.
处理法线问题时，快速画一个切线及其垂直关系的小草图。这种视觉检查通常能防止忘记翻转斜率的错误。最后，多练习IB历年真题中结合微分与切法线的问题，因为它们经常出现在A部分和B部分。
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