📚 Exercise 5E | 练习5E
This revision set focuses on the essential techniques of differentiation required in IB Mathematics, consolidating the rules and their applications through a collection of typical questions you will encounter in Exercise 5E. Mastering these skills is crucial not only for solving routine derivative problems but also for tackling tangents, normals, stationary points, and real-world contexts that appear throughout the syllabus.
本复习专题围绕IB数学中必备的微分技巧展开，通过习题5E中的典型题目，巩固各种求导法则及其应用。掌握这些技能不仅有助于解决常规的导数计算问题，也是处理切线、法线、驻点以及真实情境应用题的基础，贯穿整个课程大纲。
1. Introduction to Differentiation | 微分简介
n changes with respect to its independent variable. For a curve y = f(x), the derivative f' insight into increasing or decreasing behaviour, turning points, and the shape of graphs.
微分衡量函数相对于自变量的变化率。对于曲线 y = f(x)，导数 f'(x) 表示任意一点处切线的斜率，能帮助我们判断函数的增减性、极值点以及图像的形状特征。
In IB Mathematics, you are expected to differentiate polynomials, trigonometric, exponential, and logarithmic functions, and to combine them using the product, quotient, and chain rules. Exercise 5E typically practises the routine application of these rules before moving to more complex contexts.
在IB数学中，你需要能够对多项式函数、三角函数、指数函数和对数函数求导，并能熟练运用乘积法则、商法则和链式法则。习题5E通常通过基础练习让你熟悉这些规则，为后续复杂背景中的应用做好准备。
2. Basic Differentiation Rules | 基本微分法则
The foundation of differentiation is the power rule: if f(x) = xⁿ, then f' erm. For example, the derivative of 5x⁴ is 20x³, and the derivative of a constant is zero.
微分的基础是幂函数法则：如果 f(x) = xⁿ，那么 f'(x) = n xⁿ⁻¹。结合常数倍法则与和法则，就可以逐项对任何多项式求导。例如，5x⁴ 的导数是 20x³，而常数的导数为0。
Common basic results are summarised in the table below. Make sure you can state these from memory, as they will be used repeatedly in more advanced differentiation tasks.
下面的表格总结了常见的基本求导公式。请务必熟记这些结果，因为在更复杂的微分运算中它们会被反复使用。
| f(x) | f'(x) |
| k (constant) | 0 |
| xⁿ | n xⁿ⁻¹ |
| a f(x) | a f'(x) |
| f(x) + g(x) | f'(x) + g'(x) |
3. The Product Rule | 乘积法则
e product of the derivatives. Instead, the product rule states: if y = u v, then dy/dx = u' v + u v' s essential to identify which part to call u and which to call v before applying the rule.
当对两个函数的乘积 u(x) · v(x) 求导时，导数不是各自导数的简单相乘。乘积法则指出：若 y = u v，则 dy/dx = u' v + u v'。在使用该法则之前，清晰地选定哪个部分作为 u、哪个作为 v 非常关键。
For instance, differentiate y = x² sin x. Set u = x² and v = sin x. Then u' = 2x, v' ng gives dy/dx = 2x·sin x + x²·cos x. Always simplify the final expression where possible.
例如，对 y = x² sin x 求导。令 u = x²，v = sin x。则 u' = 2x，v' = cos x。代入公式得 dy/dx = 2x·sin x + x²·cos x。最后尽可能化简结果。
4. The Quotient Rule | 商法则
For a function expressed as a quotient u(x) / v(x), the quotient rule is used: dy/dx = (u' v − u v') / v². A common mnemonic is 'low d-high minus high d-low over low squared'. Take care with the subtraction order, as it affects the sign of the derivative.
对于写成商的形式 u(x) / v(x) 的函数，需要使用商法则：dy/dx = (u' v − u v') / v²。一个常见的记忆方法是“分母导数乘分子减分子导数乘分母，除以分母的平方”。务必注意减法的顺序，因为顺序会影响到导数的符号。
Differentiate y = (x² + 1) / cos x. Here u = x² + 1, v = cos x. Then u' = 2x and v' is dy/dx = (2x·cos x − (x² + 1)·(−sin x)) / cos² x = (2x cos x + (x² + 1) sin x) / cos² x.
对 y = (x² + 1) / cos x 求导。此处 u = x² + 1，v = cos x。那么 u' = 2x，v' = −sin x。导数计算为 dy/dx = (2x·cos x − (x² + 1)·(−sin x)) / cos² x = (2x cos x + (x² + 1) sin x) / cos² x。
5. The Chain Rule | 链式法则
composite functions, where one function is inside another. If y = f(g(x)), then dy/dx = f'(g(x)) · g' (dy/du) × (du/dx). This rule enables you to differentiate expressions like (3x² − 5)⁸ or esin x.
链式法则用于处理复合函数，即一个函数嵌套在另一个函数内部的情形。若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。用莱布尼茨记号表示为 dy/dx = (dy/du) × (du/dx)。利用这一法则，你可以对 (3x² − 5)⁸ 或 esin x 这类表达式求导。
To differentiate y = √4x + 1, rewrite as y = (4x + 1)½. The outer function is u½ and the inner is u = 4x + 1. Applying the chain rule gives dy/dx = ½ (4x + 1)−½ × 4 = 2 / √4x + 1. Practice with different inner functions to build speed.
对 y = √4x + 1 求导，首先将其改写为 y = (4x + 1)½。外层函数是 u½，内层为 u = 4x + 1。应用链式法则得到 dy/dx = ½ (4x + 1)−½ × 4 = 2 / √4x + 1。通过更换不同的内层函数进行练习，可以提升运算速度。
6. Differentiating Trigonometric Functions | 三角函数的微分
hen angles are multiples of x, such as sin(kx) or cos(ax + b), the chain rule must be applied.
三角函数的导数在IB题目中出现频率很高。核心结论为：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。当角度是 x 的倍数时，比如 sin(kx) 或 cos(ax + b)，必须配合链式法则求导。
orth memorising the derivatives of sec x, csc x, and cot x if your syllabus includes them.
对 y = sin(5x) 求导，得到 dy/dx = cos(5x) × 5 = 5 cos(5x)。类似地，对于 y = tan(2x − π/4)，可得 dy/dx = sec²(2x − π/4) × 2。如果你的课程大纲要求，也值得记住 sec x、csc x 和 cot x 的导数。
7. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The natural exponential function eˣ is unique because its derivative is itself: d/dx (eˣ) = eˣ. For y = e^(f(x)), the chain rule gives dy/dx = f' ifferentiates as d/dx (ln x) = 1/x, and the chain rule extends this to d/dx (ln(f(x))) = f'(x)/f(x).
自然指数函数 eˣ 的特殊之处在于其导数就是它本身：d/dx (eˣ) = eˣ。对于 y = e^(f(x))，链式法则给出 dy/dx = f'(x) e^(f(x))。自然对数函数的导数为 d/dx (ln x) = 1/x，结合链式法则可推广为 d/dx (ln(f(x))) = f'(x)/f(x)。
For instance, differentiate y = e3x² − x. Here f(x) = 3x² − x, so f'(x) = 6x − 1. Hence dy/dx = (6x − 1) e3x² − x. For logarithmic functions, y = ln(cos x) gives dy/dx = −sin x / cos x = −tan x.
例如，对 y = e3x² − x 求导。此时 f(x) = 3x² − x，f'(x) = 6x − 1，于是 dy/dx = (6x − 1) e3x² − x。对于对数函数，y = ln(cos x) 的导数是 dy/dx = −sin x / cos x = −tan x。
8. Tangents and Normals | 切线与法线
Once you can find f' f the tangent to y = f(x) at a given point (a, f(a)). The gradient of the tangent is m = f' − a). The normal is perpendicular to the tangent, so its gradient is −1/m, provided m ≠ 0.
一旦能求出 f'(x)，就可以确定曲线 y = f(x) 上给定点 (a, f(a)) 处的切线方程。切线的斜率为 m = f'(a)，其方程为 y − f(a) = m (x − a)。法线与切线垂直，因此法线的斜率为 −1/m（只要 m ≠ 0）。
For y = x³ − 2x at x = 1, the point is (1, −1) and f' dient −1 and equation y + 1 = −1(x − 1), i.e. y = −x. Practice finding both forms quickly.
对于 y = x³ − 2x 在 x = 1 处，点为 (1, −1)，f'(1) = 3(1)² − 2 = 1。切线方程为 y + 1 = 1(x − 1)，即 y = x − 2。法线的斜率为 −1，方程为 y + 1 = −1(x − 1)，即 y = −x。请多加练习，以便快速求出这两种直线方程。
9. Stationary Points and Turning Points | 驻点与极值点
Stationary points occur where f' cal maxima, local minima, or points of inflection. To classify them, examine the sign of f' oint. A change from positive to negative indicates a maximum, while a change from negative to positive indicates a minimum.
驻点出现在 f'(x) = 0 的位置。它们可能是局部最大值、局部最小值或拐点。为了判断类型，需要检查驻点左右两侧 f'(x) 的符号。从正变负说明是极大值点，从负变正则说明是极小值点。
For f(x) = x³ − 3x, f'(x) = 3x² − 3 = 3(x − 1)(x + 1). Setting f' (−1, 2) and a minimum at (1, −2). Always state coordinates when asked for turning points.
对于 f(x) = x³ − 3x，f'(x) = 3x² − 3 = 3(x − 1)(x + 1)。令 f'(x) = 0 得 x = −1 和 x = 1。通过区间测试可知，x = −1 处为极大值点 (−1, 2)，x = 1 处为极小值点 (1, −2)。题目若要求给出极值点，务必写出坐标。
10. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative, f''(x), is the derivative of f'(x). It describes the concavity of the graph: if f''(x) > 0, the curve is concave up; if f'' e down. A point of inflection occurs where concavity changes, which often coincides with f''(x) = 0, but you must verify the sign change.
二阶导数 f''(x) 是 f'(x) 的导数，它描述了函数图像的凹凸性：若 f''(x) > 0，曲线是凹向上的；若 f''(x) < 0，曲线是凹向下的。凹凸性发生改变的点称为拐点，往往出现在 f''(x) = 0 处，但必须验证符号是否确实改变。
Using the second derivative can also classify stationary points more efficiently. If f'(a) = 0 and f''(a) < 0, the point is a maximum; if f''(a) > 0, it is a minimum. When f''(a) = 0, the test is inconclusive, and you should use the first derivative test instead.
利用二阶导数还能更高效地判断驻点类型。若 f'(a) = 0 且 f''(a) < 0，该点为极大值点；若 f''(a) > 0，则为极小值点。当 f''(a) = 0 时，二阶导数判别法失效，此时应改用一阶导数符号测试。
11. Applied Differentiation Problems | 微分应用题
xtual problems where differentiation helps optimise a variable, such as maximising area, minimising surface area, or finding rates of change. These questions require you to form a function in one variable, differentiate it, and then set f' lues. Always check the nature of the stationary point and interpret it within the context.
IB考试中经常出现应用题，要求利用微分优化某个变量，例如最大化面积、最小化表面积或求变化率。这类问题需要你先建立关于单一变量的函数，接着求导并令 f'(x) = 0 以找到最优值。务必检验驻点的性质，并结合实际背景进行解释。
tive confirms a maximum volume of about 592.6 cm³. Clearly state the domain (0 < x < 10) and justify your answer.
例如，用一块边长为20 cm的正方形纸板，在四角剪去小正方形做成无盖盒：体积 V = x(20 − 2x)²。展开并求导后，令 dV/dx = 0 得到 x = 10/3。二阶导数证明确为最大体积，约为592.6 cm³。务必注明定义域 0 < x < 10 并验证答案。
12. Tips for Exercise 5E | 练习5E的解题技巧
Begin by carefully reading the function to identify which differentiation rule applies. If you spot a product, quotient, or composition, label u and v immediately. It is safer to simplify the expression before differentiating whenever possible – for example, divide a simple denominator into the numerator rather than using the quotient rule.
解题时先仔细审读函数，确定适用的微分法则。一旦看到乘积、商或者复合结构，应立即标注 u 和 v。尽可能在求导前先化简表达式——例如，当分母较简单时，可用除法拆分为各项分别求导，而不要盲目套用商法则。
After differentiating, always check if the derivative can be factorised or simplified; this will be useful when finding stationary points or tangents. Finally, watch out for common mistakes like forgetting the negative sign from the derivative of cos x, misapplying the chain rule, or dropping brackets in the quotient rule. Thorough practice with Exercise 5E will solidify these skills.
求导后，务必检查导数能否因式分解或化简，这对后续求驻点或切线极为有用。最后，留心常见错误，比如漏掉 cos x 导数的负号、用错链式法则，或在商法则中去掉了括号。通过习题5E的充分练习，这些技能将得到巩固。
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