📚 Rules of differentiation | 微分法则
Differentiation is a fundamental tool in calculus that allows us to find the instantaneous rate of change of a function. In IB Mathematics, mastering the rules of differentiation — including the power rule, product rule, quotient rule, and chain rule — is essential for tackling a wide range of problems, from curve sketching to optimisation. These rules form the backbone of calculus and provide a systematic way to differentiate polynomial, trigonometric, exponential, and logarithmic functions without relying on first principles every time.
微分是微积分中的一个基本工具，用于求函数的瞬时变化率。在 IB 数学中，掌握各种微分法则（包括幂法则、积法则、商法则和链式法则）是处理从曲线描绘到最优化等各种问题的关键。这些法则构成了微积分的主干，提供了一种系统的方法来对多项式、三角函数、指数函数和对数函数求导，而无需每次都依赖第一原理。
1. The Power Rule | 幂法则
The power rule is the most straightforward differentiation rule. If f(x) = xn where n is any real number, then f'(x) = n xn-1. This rule works for positive integers, negative integers, fractions, and irrational exponents. For example, differentiating x5 gives 5x4, while the derivative of x½ (i.e. √x) is ½ x-½.
幂法则是最直接的求导法则。若 f(x) = xn，其中 n 为任意实数，则 f'(x) = n xn-1。此法则适用于正整数、负整数、分数以及无理指数。例如，对 x5 求导得 5x4，而 x½（即 √x）的导数为 ½ x-½。
Negative exponents follow the same pattern: the derivative of x-3 is -3x-4. For rational exponents like x⅔, the derivative becomes ⅔ x-⅓. Always rewrite roots as fractional powers before differentiating to apply the power rule easily.
负指数也遵循相同的模式：x-3 的导数为 -3x-4。对于有理指数，如 x⅔，其导数变为 ⅔ x-⅓。在求导之前，务必将根式重写为分数次幂，以便轻松应用幂法则。
2. Constant Multiple and Sum/Difference Rules | 常数倍法则与和差法则
If k is a constant, the derivative of k f(x) is simply k f' he sum of the derivatives, we can differentiate any polynomial term by term: (f(x) + g(x))' = f'(x) + g'(x). The difference rule works identically: (f(x) - g(x))' = f'(x) - g'(x).
若 k 为常数，则 k f(x) 的导数就是 k f'(x)，这就是常数倍法则。结合和法则，即和的导数等于导数之和，我们可以逐项对任意多项式求导：(f(x) + g(x))' = f'(x) + g'(x)。差法则与之完全相同：(f(x) - g(x))' = f'(x) - g'(x)。
For example, to differentiate 3x4 - 2x3 + 5x - 7, you take the derivative of each term separately: 12x3 - 6x2 + 5. The constant term -7 becomes 0 because the derivative of a constant is always zero.
例如，要对 3x4 - 2x3 + 5x - 7 求导，只需分别对每一项求导：12x3 - 6x2 + 5。常数项 -7 变为 0，因为常数的导数始终为零。
3. The Product Rule | 积法则
ot simply multiply their derivatives. The product rule states: if f(x) = u(x) v(x), then f'(x) = u'(x) v(x) + u(x) v' second, plus first times derivative of second. This rule is essential for functions like x2 sin x or ex ln x.
在对两个函数的乘积求导时，不能简单地将它们的导数相乘。积法则指出：若 f(x) = u(x) v(x)，则 f'(x) = u'(x) v(x) + u(x) v'(x)。用语言表达就是：第一项的导数乘以第二项，再加上第一项乘以第二项的导数。这一法则对于诸如 x2 sin x 或 ex ln x 之类的函数至关重要。
A common example: differentiate f(x) = x3 cos x. Let u = x3 (u' = 3x2) and v = cos x (v' = -sin x). Then f'(x) = 3x2 cos x + x3 (−sin x) = 3x2 cos x - x3 sin x. It is helpful to factor out common terms where possible, but not always necessary.
一个常见的例子：对 f(x) = x3 cos x 求导。设 u = x3 (u' = 3x2)，v = cos x (v' = -sin x)。则 f'(x) = 3x2 cos x + x3 (−sin x) = 3x2 cos x - x3 sin x。在可能的情况下提取公因式会很有帮助，但这并非总是必要。
The product rule can be extended to three functions: (u v w)' = u' v w + u v' w + u v w' uently tested on the two-function version, but recognising the pattern helps avoid errors.
积法则可以扩展到三个函数：(u v w)' = u' v w + u v' w + u v w'。在 IB 考试中，最常考的是两个函数的版本，但认识这一模式有助于避免错误。
4. The Quotient Rule | 商法则
en differentiating a ratio of two functions. If f(x) = u(x) / v(x), where v(x) ≠ 0, then f'(x) = [u'(x) v(x) - u(x) v'(x)] / [v(x)]2. The numerator is "bottom times derivative of top, minus top times derivative of bottom". This rule can be remembered with the mnemonic "low d high minus high d low, square the bottom and away you go".
商法则用于对两个函数的比值求导。若 f(x) = u(x) / v(x)，且 v(x) ≠ 0，则 f'(x) = [u'(x) v(x) - u(x) v'(x)] / [v(x)]2。其分子为“分母乘以分子的导数，减去分子乘以分母的导数”。这个法则可以用口诀“低 d 高减去高 d 低，分母平方就搞定”来帮助记忆。
Consider f(x) = x2 / (x+1). Here u = x2, u' = 2x; v = x+1, v' = 1. Applying the quotient rule gives f'(x) = [2x(x+1) - x2(1)] / (x+1)2 = (2x2+2x - x2) / (x+1)2 = (x2+2x) / (x+1)2. Always check if the function can be simplified before using the rule; sometimes rewriting as x2(x+1)-1 and using the product or chain rule is easier.
考虑 f(x) = x2 / (x+1)。这里 u = x2，u' = 2x；v = x+1，v' = 1。应用商法则得 f'(x) = [2x(x+1) - x2(1)] / (x+1)2 = (2x2+2x - x2) / (x+1)2 = (x2+2x) / (x+1)2。在使用该法则之前，始终应检查函数是否可以化简；有时将其重写为 x2(x+1)-1 并使用积法则或链式法则会更容易。
5. The Chain Rule | 链式法则
omposite functions, i.e. functions of the form f(g(x)). It states that the derivative is f'(g(x)) * g' The chain rule is arguably the most important differentiation tool because it allows us to handle nested functions such as sin(x2), ecos x, or ln(3x+5).
链式法则用于对复合函数求导，即形如 f(g(x)) 的函数。该法则指出，导数为 f'(g(x)) * g'(x)。用莱布尼茨记号表示，若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du * du/dx。链式法则可以说是最重要的求导工具，因为它使得我们能够处理像 sin(x2)、ecos x 或 ln(3x+5) 这样的嵌套函数。
A classic example: differentiate y = (3x2 + 1)5. Let u = 3x2+1, then y = u5. dy/du = 5u4, du/dx = 6x. So dy/dx = 5(3x2+1)4 * 6x = 30x(3x2+1)4. Once you get comfortable, you can differentiate "outside function then multiply by the derivative of the inside" without explicitly writing u.
一个经典例子：对 y = (3x2 + 1)5 求导。令 u = 3x2+1，则 y = u5。dy/du = 5u4，du/dx = 6x。因此 dy/dx = 5(3x2+1)4 * 6x = 30x(3x2+1)4。一旦熟练之后，就可以采用“先对外层函数求导，再乘以内部函数的导数”的方法，而无需明确写出 u。
When the chain rule is combined with the product or quotient rule, careful algebraic simplification is needed. For instance, differentiating x2 (1+x3)4 requires product rule with chain rule on the second factor.
当链式法则与积法则或商法则结合使用时，需要进行仔细的代数化简。例如，对 x2 (1+x3)4 求导就需要积法则，并对第二个因式使用链式法则。
6. Derivatives of Exponential Functions | 指数函数的导数
The natural exponential function ex is unique because its derivative is itself: d/dx (ex) = ex. More generally, for any positive constant a, the derivative of ax is ax ln a. Thus d/dx (2x) = 2x ln 2, and d/dx (e3x) = e3x * 3 = 3e3x nstant e ≈ 2.71828 is the only base for which the derivative is exactly the same function.
自然指数函数 ex 是独一无二的，因为它的导数就是它本身：d/dx (ex) = ex。更一般地，对于任何正常数 a，ax 的导数为 ax ln a。因此 d/dx (2x) = 2x ln 2，而 d/dx (e3x) = e3x * 3 = 3e3x（使用链式法则）。常数 e ≈ 2.71828 是唯一使得导数恰好等于原函数的底数。
When the exponent is a function of x, always apply the chain rule. For example, the derivative of ex² is ex² * 2x = 2x ex². Similarly, d/dx (5sin x) = 5sin x ln 5 * cos x. These appear frequently in both SL and HL topics.
当指数是 x 的函数时，一定要应用链式法则。例如，ex² 的导数为 ex² * 2x = 2x ex²。类似地，d/dx (5sin x) = 5sin x ln 5 * cos x。这些在 SL 和 HL 内容中都经常出现。
7. Derivatives of Logarithmic Functions | 对数函数的导数
The derivative of the natural log function ln x (x > 0) is 1/x. For logarithms with other bases, use the change-of-base formula: d/dx (loga x) = 1 / (x ln a). Thus d/dx (log10 ike ln(ax+b), the chain rule gives a/(ax+b). For example, the derivative of ln(2x+3) is 2/(2x+3).
自然对数函数 ln x（x > 0）的导数是 1/x。对于其他底数的对数，可以利用换底公式：d/dx (loga x) = 1 / (x ln a)。因此 d/dx (log10 x) = 1 / (x ln 10)。对于如 ln(ax+b) 这样的复合形式，链式法则给出 a/(ax+b) 的结果。例如，ln(2x+3) 的导数是 2/(2x+3)。
If the logarithmic argument is more complex, like ln(x2+1), differentiate the outer function and multiply by the derivative of x2+1, yielding (2x)/(x2+1). Logarithmic differentiation can simplify the differentiation of products and quotients, but that is a separate technique often used in HL.
如果对数的自变量更复杂，比如 ln(x2+1)，则对外层函数求导并乘以 x2+1 的导数，得到 (2x)/(x2+1)。对数求导法可以简化积与商的求导，但这是另一种技巧，通常在 HL 中使用。
8. Derivatives of Trigonometric Functions | 三角函数的导数
c functions are essential: d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec2 x (or 1/cos2 x). It is crucial to remember the negative sign in the derivative of cosine. The derivative of tan x can be derived from sin x/cos x using the quotient rule.
三个基本三角函数的导数是必须掌握的：d/dx (sin x) = cos x，d/dx (cos x) = -sin x，d/dx (tan x) = sec2 x（或 1/cos2 x）。关键在于要记住余弦的导数带有负号。tan x 的导数可以借助商法则从 sin x/cos x 推导出来。
chain rule covers all standard forms. For instance, d/dx (sin 5x) = 5 cos 5x; d/dx (cos(3x2+1)) = -sin(3x2+1) * 6x = -6x sin(3x2+1). The identity sec2 x = 1 + tan2 x sometimes helps with simplification. IB students are not required to know derivatives of csc, sec, and cot, but HL may include them via the reciprocal identities and quotient rule.
将这些与链式法则结合就能涵盖所有标准形式。例如，d/dx (sin 5x) = 5 cos 5x；d/dx (cos(3x2+1)) = -sin(3x2+1) * 6x = -6x sin(3x2+1)。恒等式 sec2 x = 1 + tan2 x 有时有助于化简。IB 学生不需要知道 csc、sec 和 cot 的导数，但 HL 可能会借助倒数恒等式和商法则涉及这些内容。
9. Higher-Order Derivatives | 高阶导数
The derivative obtained by differentiating a function once is the first derivative, denoted f'(x) or dy/dx. Differentiating again gives the second derivative f''(x) or d2y/dx2. The second derivative measures the rate of change of the slope and is used to determine concavity and points of inflection. Higher-order derivatives like f'''(x) follow the same pattern.
对函数求一次导后得到的导数是为一阶导数，记作 f'(x) 或 dy/dx。再次求导便得到二阶导数 f''(x) 或 d2y/dx2。二阶导数衡量的是斜率的变化率，用于判断函数的凹凸性和拐点。更高阶的导数，如 f'''(x)，遵循同样的模式。
For example, if f(x) = x4, then f'(x) = 4x3, f''(x) = 12x2, f''' x is again sin x. Higher derivatives appear in kinematics: position s(t), velocity v(t)=s'(t), acceleration a(t)=v'(t)=s''(t), and jerk j(t)=s'''(t).
例如，若 f(x) = x4，则 f'(x) = 4x3，f''(x) = 12x2，f'''(x) = 24x。对于三角函数，导数的模式每四次重复一次。sin x 的四阶导数又回到了 sin x。高阶导数在运动学中出现：位置 s(t)、速度 v(t)=s'(t)、加速度 a(t)=v'(t)=s''(t)，以及加加速度 j(t)=s'''(t)。
10. Differentiation of Special Functions | 特殊函数的导数
Beyond standard elementary functions, IB students may encounter implicit differentiation where equations define y implicitly as a function of x. For such cases, chain rule terms like dy/dx appear when differentiating y with respect to x. This technique is vital for handling curves like x2 + y2 = 25.
除了标准初等函数之外，IB 学生可能会遇到隐函数求导，即方程隐式地将 y 定义为 x 的函数。对于这种情况，当对 y 关于 x 求导时，会出现像 dy/dx 这样的链式法则项。这种方法对于处理诸如 x2 + y2 = 25 之类的曲线至关重要。
al case is differentiation of inverse trigonometric functions such as d/dx (arcsin x) = 1/√1 - x2, but these are mainly HL content and appear on the formula booklet. Always consult the IB formula booklet for less common derivatives.
对两边关于 x 求导得 2x + 2y * dy/dx = 0，因此 dy/dx = -x/y。另一种特殊情况是反三角函数的求导，例如 d/dx (arcsin x) = 1/√1 - x2，但这主要属于 HL 内容，并会在公式手册中给出。对于不太常见的导数，始终应查阅 IB 公式手册。
11. Applying Differentiation Rules – Tangents and Normals | 应用微分法则 – 切线与法线
gent line to a curve at a given point. The derivative at point x = a gives the slope m = f'(a). The tangent line then has equation y - f(a) = m (x - a). For example, for y = x3 - 2x at x=1, y(1) = -1, m = 3(1)2 - 2 = 1, so the tangent is y + 1 = 1(x - 1), i.e. y = x - 2.
微分法则最常见的应用之一是求曲线在给定点处的切线方程。在 x = a 处的导数值给出了斜率 m = f'(a)。切线方程进而为 y - f(a) = m (x - a)。例如，对于 y = x3 - 2x，在 x=1 处，y(1) = -1，m = 3(1)2 - 2 = 1，因此切线方程为 y + 1 = 1(x - 1)，即 y = x - 2。
y = -x. These geometric applications reinforce the need to compute derivatives accurately.
法线垂直于切线，因此其斜率为 -1/m（只要 m ≠ 0）。沿用同一个例子，法线的斜率为 -1，方程为 y + 1 = -1(x - 1)，即 y = -x。这些几何应用强化了精确计算导数的必要性。
12. Summary and Common Pitfalls | 总结与常见错误
To summarise, the main differentiation rules are: power rule (xn → n xn-1), sum/difference, product rule, quotient rule, and the chain rule. Combine these with formulas for exponential, logarithmic, and trigonometric functions to differentiate almost any function encountered at IB level. Remember that not all functions can be simplified first, and rewriting expressions in index form often saves time.
总之，主要的微分法则有：幂法则 (xn → n xn-1)、和/差法则、积法则、商法则以及链式法则。将这些与指数函数、对数函数和三角函数的求导公式相结合，就能对 IB 阶段遇到的几乎所有函数进行求导。记住，并非所有函数都能先化简，将表达式重写为指数形式常常可以节省时间。
Common mistakes include forgetting the inner derivative when using the chain rule, misplacing terms in the quotient rule numerator, and applying the product rule incorrectly when more than two factors are present. Always check the domain of the original function and its derivative; for example, ln x is defined only for x>0. Practice with a variety of functions will build fluency and confidence for the exam.
常见的错误包括：在使用链式法则时忘记乘以内层函数的导数；商法则分子中的项放错位置；以及在处理超过两个因式的乘积时错误地应用积法则。要始终检查原函数及其导数的定义域；例如，ln x 仅在 x>0 时有定义。通过针对各种函数进行练习，可以提升熟练度与应试信心。
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