📚 IB Math Review Set 7C: Mastering Differentiation | IB数学复习题集7C：精通微分
Differentiation is a cornerstone of the IB Mathematics syllabus, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI) courses. This revision set revisits the essential techniques and applications you need to solve problems involving rates of change, curve sketching, optimisation, and more. Whether you are working through textbook exercises or past paper questions, a solid command of derivative rules and their geometric meaning will help you tackle Review Set 7C with confidence.
微分是IB数学大纲的基石，无论是分析与方法(AA)还是应用与解释(AI)课程都占有重要地位。本复习集回顾必要的技巧和应用，帮助你解决变化率、曲线草图、最优化等问题。无论你在做课本习题还是历年真题，扎实掌握导数法则及其几何意义，都能让你自信应对复习题集7C。
1. Power Rule and Basic Derivatives | 幂法则与基本导数
nge. The simplest rule is the power rule: if f(x) = xⁿ, where n is any real number, then f' tiple rule and the sum/difference rule, you can differentiate any polynomial term by term.
函数的导数给出瞬时变化率。最简单的法则是幂法则：若 f(x) = xⁿ（n为任意实数），则 f'(x) = n xⁿ⁻¹。该法则适用于正指数、负指数和分数指数。结合常数倍法则与和差法则，你可以对多项式的每一项逐项求导。
d/dx (xⁿ) = n xⁿ⁻¹
For a constant c, d/dx (c) = 0. For a sum: d/dx [u(x) ± v(x)] = u'(x) ± v'(x). A common mistake is forgetting to rewrite terms like 1/x as x⁻¹ or √x as x½ before differentiating.
常数 c 的导数为 0。和差法则：d/dx [u(x) ± v(x)] = u'(x) ± v'(x)。常见错误是在求导前忘记将 1/x 改写为 x⁻¹，或将 √x 改写为 x½。
| Function f(x) | Derivative f'(x) |
|---|---|
| c (constant) | 0 |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
Mastering these basic forms allows you to quickly handle the functions encountered in Review Set 7C without hesitation.
掌握这些基本形式后，你就能毫不犹豫地快速处理复习题集7C中遇到的各类函数。
2. Product Rule | 积法则
) v(x), the derivative is not simply the product of the derivatives. The product rule states that you take the derivative of the first function times the second, plus the first times the derivative of the second.
当一个函数是两个可微函数的乘积，例如 y = u(x) v(x)，其导数并不是简单的导数相乘。积法则要求用第一个函数的导数乘以第二个函数，再加上第一个函数乘以第二个函数的导数。
d/dx [u(x) v(x)] = u'(x) v(x) + u(x) v'(x)
rly before differentiating. For example, given y = x² sin x, let u = x², v = sin x; then u' = 2x, v' = cos x, so dy/dx = 2x sin x + x² cos x. Watch out for products of three functions – you can apply the product rule twice.
用莱布尼茨记号表示为：若 y = u v，则 dy/dx = u (dv/dx) + v (du/dx)。在求导前要明确设定 u 和 v。例如，对于 y = x² sin x，设 u = x²，v = sin x；则 u' = 2x，v' = cos x，因此 dy/dx = 2x sin x + x² cos x。注意三个函数相乘的情形——可以两次应用积法则。
3. Quotient Rule | 商法则
u(x)/v(x), where v(x) ≠ 0, the derivative is given by the quotient rule. The formula may look intimidating, but remembering it as “low d-high minus high d-low over low squared” can help.
对于形如 y = u(x)/v(x) 的有理函数，其中 v(x) ≠ 0，其导数由商法则给出。公式看起来可能有点吓人，但记住“分母乘分子的导数减去分子乘分母的导数，再除以分母的平方”会很有帮助。
d/dx [u(x)/v(x)] = (u'(x) v(x) − u(x) v'(x)) / [v(x)]²
cel terms. For instance, differentiate y = (x² + 1)/(x − 1). Set u = x²+1, v = x−1, then u'=2x, v' −1 = x²−2x−1. This expression can be left as is or further analysed for stationary points.
IB考试技巧：使用法则后务必简化分子，也许可以因式分解并约分。例如，对 y = (x² + 1)/(x − 1) 求导。设 u = x²+1, v = x−1，则 u'=2x, v'=1。导数为 [(2x)(x−1) − (x²+1)(1)]/(x−1)²。展开分子得 2x²−2x−x²−1 = x²−2x−1。结果可以保留此形式，或进一步用于研究驻点。
4. Chain Rule | 链式法则
), let u = g(x) so that y = f(u). Then dy/dx = dy/du × du/dx. In prime notation, [f(g(x))]' = f'(g(x)) g'(x).
链式法则用于对复合函数（即函数的函数）求导。若 y = f(g(x))，令 u = g(x) 使 y = f(u)，则 dy/dx = dy/du × du/dx。用带撇记号表示为 [f(g(x))]' = f'(g(x)) g'(x)。
dy/dx = f'(g(x)) g'(x)
This rule is indispensable for functions like y = (3x + 2)⁵, y = sin(2x), or y = ex² u = 3x+2; then y = u⁵, dy/du = 5u⁴, du/dx = 3. Multiplying gives dy/dx = 15(3x+2)⁴. Always “multiply by the derivative of the inside function” udents often forget this final step, especially in trigonometric or logarithmic functions.
该法则在处理 y = (3x + 2)⁵, y = sin(2x) 或 y = ex² 等函数时不可或缺。对于 y = (3x+2)⁵，令 u = 3x+2；则 y = u⁵, dy/du = 5u⁴, du/dx = 3。相乘得 dy/dx = 15(3x+2)⁴。必须始终“乘以内层函数的导数”。学生们常忘记这最后一步，尤其是在三角函数或对数函数中。
5. Tangent and Normal Lines | 切线与法线方程
The derivative at a point x = a, f' t point. The equation of the tangent can then be written in point-slope form: y − f(a) = f'(a)(x − a). The normal line is perpendicular to the tangent, so its slope mnormal = −1 / f'(a), provided f'(a) ≠ 0.
函数在 x = a 处的导数 f'(a) 给出了曲线 y = f(x) 在该点切线的斜率。切线方程可写成点斜式：y − f(a) = f'(a)(x − a)。法线垂直于切线，因此其斜率 mnormal = −1 / f'(a)，前提是 f'(a) ≠ 0。
m_tangent = f'(a), mnormal = −1 / f'(a)
mal, take care to use the negative reciprocal. For example, for f(x) = x³ − 3x at x = 2, f' − 3 = 9. Tangent: y − 2 = 9(x − 2). Normal: slope = −1/9, equation: y − 2 = (−1/9)(x − 2).
在典型的IB题目中，你需要先求出导数，代入 x 坐标，然后列出直线方程。若要求法线方程，注意使用负倒数。例如，对 f(x) = x³ − 3x 在 x=2 处，f'(2)=3(2)²−3=9。切线：y−2=9(x−2)。法线斜率=−1/9，方程：y−2=(−1/9)(x−2)。
6. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative, f'' the derivative of the first derivative. It tells us about the concavity of the graph. If f''(x) > 0 on an interval, the curve is concave up (like a cup). If f'' wn (like a cap). Points where concavity changes are inflection points, which occur where f''(x) = 0 or is undefined, and the sign of f'' actually changes.
二阶导数 f''(x) 或 d²y/dx²，是一阶导数的导数。它反映了函数图像的凹凸性。若在某个区间内 f''(x) > 0，曲线向上凹（像杯子）；若 f''(x) < 0，则向下凹（像帽子）。凹凸发生变化的点叫做拐点，出现在 f''(x) = 0 或无定义，且 f'' 的符号确实发生变化之处。
Concavity: f''(x) > 0 ⇒ concave up; f''(x) < 0 ⇒ concave down
to IB questions often ask you to determine intervals of concavity and locate points of inflection. To do this, compute f''(x), set it to zero, solve for x, and test intervals. For f(x) = x³ − 3x, f''(x) = 6x, zero at x=0. For x<0, f'' ive (concave down); for x>0, positive (concave up). Hence, (0,0) is a point of inflection.
IB题目常常要求确定凹凸区间并找出拐点。为此需要计算 f''(x)，令其等于0，解出 x，并检验区间。对于 f(x)=x³−3x，f''(x)=6x，在 x=0 处为零。x<0 时 f'' 为负（向下凹），x>0 时为正（向上凹）。因此 (0,0) 是拐点。
7. Stationary Points and Optimization | 驻点与最优化
Stationary points occur where f' ints of inflection. The second derivative test provides a quick way to classify them: if f'(a)=0 and f''(a)>0, then x=a is a local minimum; if f''(a)<0, it is a local maximum. If f''(a)=0, the test is inconclusive, and you must examine the sign of f' around a.
驻点出现在 f'(x) = 0 处。它们可以是局部极大值、局部极小值或驻点拐点。二阶导数测试提供了一种快捷的分类方法：若 f'(a)=0 且 f''(a)>0，则 x=a 处为局部极小值；若 f''(a)<0，则为局部极大值。若 f''(a)=0，则该测试失效，必须考察 f' 在 a 附近的符号。
Optimisation: find f'(x)=0, then use f''(x) or a sign diagram to classify.
Optimisation problems involve finding maximum or minimum values of a quantity that can be modelled by a function. Start by writing an equation for the quantity in terms of one variable, differentiate, find stationary points, and check endpoints if the domain is closed. In IB, such problems often relate to areas, volumes, distances, or profit functions.
最优化问题涉及寻找可用函数建模的某个量的最大值或最小值。首先用单个变量表示该量，求导，找到驻点，并在闭区间情况下检查端点。在IB中，这类问题通常与面积、体积、距离或利润函数相关。
8. Trigonometric Functions Differentiation | 三角函数微分
Knowing the derivatives of the basic trigonometric functions is essential. The derivative of sin x is cos x; the derivative of cos x is −sin x; and the derivative of tan x is sec² x, which can also be written as 1/cos² x. These must be memorised, as they appear directly in many differentiation problems.
熟悉基本三角函数的导数是必需的。sin x 的导数是 cos x；cos x 的导数是 −sin x；tan x 的导数是 sec² x，也可写作 1/cos² x。这些必须熟记，因为它们在许多求导问题中直接出现。
d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x
the derivative of the angle. Also, be comfortable differentiating expressions like x sin x (product rule) and (sin x)/x (quotient rule).
当与链式法则结合时，这些公式可推广到 sin(kx) 和 cos(kx+b) 等函数。例如，d/dx [sin(2x)] = 2 cos(2x)，d/dx [cos(πx)] = −π sin(πx)。记住要乘以角度的导数。同时也要熟练地使用积法则和商法则处理诸如 x sin x 和 (sin x)/x 的求导。
9. Exponential and Logarithmic Functions | 指数与对数函数微分
The natural exponential function eˣ is its own derivative: d/dx (eˣ) = eˣ. For a general exponential aˣ, d/dx (aˣ) = aˣ 1/x: d/dx (ln x) = 1/x, for x>0. When the argument is more complex, apply the chain rule.
自然指数函数 eˣ 的导数就是它本身：d/dx (eˣ) = eˣ。对于一般的指数函数 aˣ，d/dx (aˣ) = aˣ ln a。自然对数函数的导数为 1/x：d/dx (ln x) = 1/x（x>0）。当自变量更复杂时，需应用链式法则。
d/dx (e^(g(x))) = e^(g(x)) g'(x), d/dx (ln(g(x))) = g'(x)/g(x)
For example, differentiate y = e2x+1. Let u = 2x+1, then dy/dx = eu × 2 = 2e2x+1 x²+1), dy/dx = (2x)/(x²+1). In IB, you may also need to differentiate expressions like x eˣ ortunities to simplify using logarithm laws before differentiating, e.g., ln(x²) = 2 ln x.
例如，对 y = e2x+1 求导。令 u = 2x+1，则 dy/dx = eu × 2 = 2e2x+1。对于 y = ln(x²+1)，dy/dx = (2x)/(x²+1)。在IB考试中，你也可能需要求 x eˣ 或 ln(sin x) 等表达式的导数。在求导前，尽量利用对数性质简化，如 ln(x²) = 2 ln x。
10. Implicit Differentiation | 隐函数微分
not explicitly given as a function of x, for example, in equations like x² + y² = 25 or ey function of x and using the chain rule: d/dx (y) = dy/dx, and d/dx (yⁿ) = n yⁿ⁻¹ (dy/dx).
隐函数微分用于 y 没有明确表示为 x 的函数的情形，例如 x² + y² = 25 或 ey + xy = 1 等方程。该方法是对方程两边同时关于 x 求导，将 y 视为 x 的函数并使用链式法则：d/dx (y) = dy/dx，以及 d/dx (yⁿ) = n yⁿ⁻¹ (dy/dx)。
d/dx (y²) = 2y (dy/dx)
estions often ask for the gradient at a specific point. Plug in the coordinates to find the numerical slope. The same technique is used in related rates problems, where two variables change with time. Always remember to multiply by dy/dx whenever differentiating a function of y.
考虑 x² + y² = 1。求导得：2x + 2y (dy/dx) = 0 → dy/dx = −x/y。IB题目常要求某一点处的斜率。代入坐标即可求得数值斜率。同样的技巧也用于相关变化率问题，即两个变量都随时间变化。切记，当对被 y 的函数求导时，必须乘以 dy/dx。
11. Summary and Common Pitfalls | 总结与常见易错点
Review Set 7C will test your ability to select and apply the correct differentiation technique. The most common mistakes include forgetting the chain rule inside trigonometric or logarithmic functions, misapplying the quotient rule sign, and mishandling negative exponents. Always rewrite radicals and fractions as powers before differentiating, and double-check the domain of logarithmic functions.
复习题集7C会考察你选择和运用正确求导技巧的能力。最常见的错误包括：在三角函数或对数函数内部
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