📚 A-Level Maths: Differentiation of Power Functions xⁿ | A-Level数学：幂函数xⁿ的求导法则
Differentiation is one of the most fundamental operations in calculus, and the power rule — the rule for differentiating functions of the form xⁿ — is the single most frequently used technique in A-Level Mathematics. Mastery of this rule is essential not only for Pure Mathematics papers but also for Mechanics, Statistics, and Further Mathematics.
微分是微积分中最基本的核心操作之一，而幂函数求导法则——即对形如 xⁿ 的函数进行求导的规则——是A-Level数学中应用最频繁的技巧。熟练掌握这一法则，不仅对 Pure Mathematics 试卷至关重要，也是 Mechanics、Statistics 乃至 Further Mathematics 的基础。
1. The Power Rule: Statement and Meaning | 幂函数法则：定义与含义
The power rule states that if y = xⁿ, where n is any real constant, then the derivative is given by dy/dx = n·xⁿ⁻¹. This means we bring the exponent down as a multiplier, and then reduce the exponent by one. For example, the derivative of x⁴ is 4x³, and the derivative of x⁻² is -2x⁻³.
幂函数法则的内容是：若 y = xⁿ，其中 n 为任意实常数，则其导数为 dy/dx = n·xⁿ⁻¹。也就是说，将指数降到前面作为系数，再将指数减一。例如，x⁴ 的导数为 4x³，x⁻² 的导数为 -2x⁻³。
If y = xⁿ, then dy/dx = n·xⁿ⁻¹
This rule holds for all real values of n, including positive integers, negative integers, fractions, and irrational numbers. The only restriction is that x must be in the domain where xⁿ⁻¹ is defined — for negative exponents, x ≠ 0.
该法则对所有实数指数 n 都成立，包括正整数、负整数、分数以及无理数。唯一需要注意的是，x 必须在 xⁿ⁻¹ 有定义的范围内——对于负指数，x 不能为 0。
2. Differentiation from First Principles | 从第一性原理推导
Although the power rule can be memorised, understanding its derivation from first principles is a key requirement of the A-Level syllabus. The derivative of f(x) = xⁿ is defined as the limit of the difference quotient as h approaches zero.
虽然幂函数法则可以直接记忆，但从第一性原理出发理解其推导过程是A-Level教学大纲中的核心要求。函数 f(x) = xⁿ 的导数定义为差商在 h 趋近于零时的极限。
f'(x) = lim(h→0) [f(x+h) – f(x)] / h
Using the binomial expansion, (x+h)ⁿ = xⁿ + n·xⁿ⁻¹h + [n(n-1)/2]·xⁿ⁻²h² + … + hⁿ. Subtracting xⁿ and dividing by h gives n·xⁿ⁻¹ + [n(n-1)/2]·xⁿ⁻²h + … + hⁿ⁻¹. As h → 0, all terms containing h vanish, leaving precisely n·xⁿ⁻¹.
利用二项式展开，(x+h)ⁿ = xⁿ + n·xⁿ⁻¹h + [n(n-1)/2]·xⁿ⁻²h² + … + hⁿ。减去 xⁿ 并除以 h 后得到 n·xⁿ⁻¹ + [n(n-1)/2]·xⁿ⁻²h + … + hⁿ⁻¹。当 h 趋近于 0 时，所有含 h 的项均消失，恰好剩下 n·xⁿ⁻¹。
3. Differentiating Positive Integer Powers | 正整数幂的求导
For positive integer values of n, the power rule is straightforward and intuitive. Each term in a polynomial of the form axⁿ can be differentiated independently: multiply by the exponent, then reduce the exponent by one.
对于正整数 n，幂函数法则非常直观。形如 axⁿ 的多项式每一项可以独立求导：先乘以指数，再将指数减一。
Consider the function y = 3x⁵. Applying the rule: dy/dx = 3 × 5x⁵⁻¹ = 15x⁴. Similarly, for y = 7x² + 4x – 9, we differentiate each term separately to obtain dy/dx = 14x + 4. The constant term -9 disappears because the derivative of any constant is zero.
考虑函数 y = 3x⁵。应用法则：dy/dx = 3 × 5x⁵⁻¹ = 15x⁴。类似地，对于 y = 7x² + 4x – 9，逐项求导得到 dy/dx = 14x + 4。常数项 -9 消失，因为任何常数的导数为 0。
Key point: when differentiating a term like 5x³, first multiply the coefficient 5 by the exponent 3 to get 15, then the new exponent is 3 – 1 = 2, giving 15x². This two-step process — multiply and reduce — is the essence of the power rule.
关键点：对 5x³ 求导时，先将系数 5 乘以指数 3 得到 15，然后新指数为 3 – 1 = 2，最终得到 15x²。这个两步过程——相乘再降幂——就是幂函数法则的精髓。
4. Negative Integer Powers | 负整数指数的求导
The power rule applies equally to negative exponents. For y = x⁻³, the derivative is dy/dx = -3x⁻⁴. It is often convenient to rewrite negative powers as fractions before differentiating, especially when the question asks for the answer in a particular form.
幂函数法则同样适用于负指数。对于 y = x⁻³，其导数为 dy/dx = -3x⁻⁴。在求导前将负指数改写为分数形式往往更方便，尤其是当题目要求以特定形式呈现答案时。
This is particularly common in mechanics problems where quantities like velocity are expressed as reciprocal powers. For example, differentiating y = 2/x² can be approached by rewriting as y = 2x⁻², then dy/dx = -4x⁻³ = -4/x³. Notice that the chain of transformations — from fraction to negative power, differentiate, then convert back — requires algebraic fluency.
这在力学问题中尤为常见，例如速度等物理量常以倒数幂的形式表达。例如，对 y = 2/x² 求导，可以先将原式改写为 y = 2x⁻²，则 dy/dx = -4x⁻³ = -4/x³。注意从分数到负指数、求导、再转换回来的变换链需要熟练的代数学技巧。
5. Fractional and Rational Powers | 分数指数与有理指数的求导
Roots and surds can be rewritten as fractional powers, allowing the power rule to be applied directly. For instance, √x = x^(1/2), and its derivative is (1/2)·x^(-1/2) = 1/(2√x). Similarly, ∛(x²) = x^(2/3), and the derivative is (2/3)·x^(-1/3).
根号和无理式可以改写为分数指数，从而直接运用幂函数法则。例如，√x = x^(1/2)，其导数为 (1/2)·x^(-1/2) = 1/(2√x)。同理，∛(x²) = x^(2/3)，导数为 (2/3)·x^(-1/3)。
When differentiating expressions containing radicals, the safest strategy is: first rewrite every root as a fractional exponent, then apply the power rule, and finally simplify the result back into radical form if required. For y = 1/√x = x^(-1/2), the derivative is dy/dx = -(1/2)x^(-3/2) = -1/(2x√x). This technique is tested regularly in A-Level papers.
对含根号的表达式求导时，最稳妥的策略是：先将所有根号改写为分数指数，然后应用幂函数法则，最后按题目要求简化为根号形式。对于 y = 1/√x = x^(-1/2)，其导数为 dy/dx = -(1/2)x^(-3/2) = -1/(2x√x)。这种技巧在A-Level试卷中经常出现。
6. Differentiating axⁿ with Constant Coefficients | 含常数系数 axⁿ 的求导
A general power function y = axⁿ, where a is a constant coefficient, has derivative dy/dx = a·n·xⁿ⁻¹. The coefficient a is simply carried along — it does not change. This is a direct consequence of the constant multiple rule in differentiation.
一般幂函数 y = axⁿ（其中 a 为常数系数）的导数为 dy/dx = a·n·xⁿ⁻¹。系数 a 只是原样保留——不发生变化。这是微分中常数倍法则的直接推论。
For example, if y = 6x⁴, then dy/dx = 6 × 4x³ = 24x³. If y = (3/4)x², then dy/dx = (3/4) × 2x = (3/2)x. This applies for any real n: y = 5x^(1/3) gives dy/dx = (5/3)x^(-2/3).
例如，若 y = 6x⁴，则 dy/dx = 6 × 4x³ = 24x³。若 y = (3/4)x²，则 dy/dx = (3/4) × 2x = (3/2)x。这适用于任何实数 n：y = 5x^(1/3) 的导数为 dy/dx = (5/3)x^(-2/3)。
Students often make the mistake of differentiating the coefficient as well, producing expressions like 24x³ from y = 6x⁴ when they accidentally treat 6 as part of the variable. Remember: only the variable term xⁿ is differentiated; the constant multiplier remains untouched until the coefficient multiplication step.
学生常犯的错误是将系数也一并求导，例如从 y = 6x⁴ 错误地得出 24x³。请记住：只有变量项 xⁿ 被求导；常数乘数在乘以指数的步骤之前保持原样。
7. Application to Polynomial Differentiation | 幂函数法则在多项式求导中的应用
Polynomials are simply sums of power functions, so the power rule combines with the sum rule to differentiate any polynomial. For a polynomial y = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, each term is differentiated independently and the results are summed.
多项式本质上是幂函数的和，因此幂函数法则与加法法则相结合就可以对任意多项式求导。对于多项式 y = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀，逐项独立求导后再将结果相加。
Example: Find the derivative of y = 3x⁴ – 5x³ + 2x² – 7x + 11.
示例：求 y = 3x⁴ – 5x³ + 2x² – 7x + 11 的导数。
dy/dx = 12x³ – 15x² + 4x – 7
Notice how the constant term 11 vanishes, and each exponent decreases by exactly one. This process is reversible through integration, which will be covered later in the course. In exam questions, polynomial differentiation is frequently the first step in finding stationary points, gradients of tangents, and rates of change.
注意常数项 11 消失，每个指数都恰好减少 1。这个过程可以通过积分来逆运算，课程稍后会讲到。在考试题目中，多项式求导通常是求驻点、切线斜率和变化率的第一步。
8. Using the Chain Rule with Power Functions | 链式法则与幂函数的结合
When a power function is composed with another function — such as y = (2x + 1)⁵ — the power rule must be applied together with the chain rule. The rule states: dy/dx = n·[f(x)]ⁿ⁻¹ · f'(x). In other words, differentiate the outer power first, then multiply by the derivative of the inner function.
当幂函数与另一个函数复合时——例如 y = (2x + 1)⁵——幂函数法则必须与链式法则结合使用。该法则指出：dy/dx = n·[f(x)]ⁿ⁻¹ · f'(x)。换句话说，先对外层幂求导，再乘以内层函数的导数。
For y = (3x² + 2)⁴, let u = 3x² + 2, then y = u⁴. We have du/dx = 6x and dy/du = 4u³. Therefore dy/dx = dy/du × du/dx = 4(3x² + 2)³ × 6x = 24x(3x² + 2)³.
对于 y = (3x² + 2)⁴，设 u = 3x² + 2，则 y = u⁴。我们有 du/dx = 6x，dy/du = 4u³。因此 dy/dx = dy/du × du/dx = 4(3x² + 2)³ × 6x = 24x(3x² + 2)³。
This is one of the most heavily examined combinations in A-Level mathematics. Students should practice recognising when the chain rule is required — typically when the base is not simply x but a more complex expression in x.
这是A-Level数学中考查频率最高的组合之一。学生应练习识别何时需要链式法则——通常当底数不是简单的 x 而是 x 的更复杂表达式时就需要使用。
9. Implicit Differentiation for Powers of y | 对 y 的幂进行隐函数求导
In equations like x² + y² = 25, the power rule must be applied carefully when differentiating terms involving y. Since y is a function of x, the derivative of yⁿ with respect to x is n·yⁿ⁻¹ · (dy/dx) — the chain rule again, where the inner function is y.
在类似 x² + y² = 25 的方程中，对含 y 的项求导时需要谨慎使用幂函数法则。因为 y 是 x 的函数，yⁿ 对 x 的导数为 n·yⁿ⁻¹ · (dy/dx)——这里再次用到链式法则，其中内层函数是 y。
Differentiating x² + y² = 25 term by term: d/dx(x²) = 2x, and d/dx(y²) = 2y·(dy/dx). Setting the derivative of the constant 25 to zero gives 2x + 2y·(dy/dx) = 0, hence dy/dx = -x/y. This technique is essential for finding gradients of curves that cannot be expressed as simple functions y = f(x).
对 x² + y² = 25 逐项求导：d/dx(x²) = 2x，d/dx(y²) = 2y·(dy/dx)。常数 25 的导数为零，因此得到 2x + 2y·(dy/dx) = 0，从而 dy/dx = -x/y。这一技巧对于求无法表示为简单函数 y = f(x) 的曲线斜率必不可少。
10. Common Pitfalls and Exam Tips | 常见错误与考试技巧
One frequent error is applying the power rule to exponential functions. The derivative of 2ˣ is not x·2ˣ⁻¹ — the power rule applies only when the variable is in the base, not in the exponent. Exponential functions require their own differentiation rules, which are covered separately.
一个常见错误是将幂函数法则误用于指数函数。2ˣ 的导数不是 x·2ˣ⁻¹——幂函数法则仅适用于底数为变量、指数为常数的情况。当变量在指数位置时，需要使用专门的指数函数求导法则。
Another common mistake is forgetting to differentiate the constant term — the derivative of any constant is always zero. Additionally, when differentiating terms like 1/x or √x, students should first rewrite them as x⁻¹ and x^(1/2) respectively to avoid sign errors and fractional exponent errors.
另一个常见错误是忘记对常数项求导——任何常数的导数恒为零。此外，对 1/x 或 √x 这样的项求导时，应先改写为 x⁻¹ 和 x^(1/2)，以避免符号错误和分数指数错误。
Examiners often award method marks for correctly applying the two-step process: multiply by the exponent and subtract one. Even if the final simplification is incorrect, showing these steps clearly can earn partial credit. Always check whether the exponent is positive, negative, or fractional, and write intermediate results in the same form as the original expression.
考官通常会对正确的两步过程给出方法分：乘以指数再减一。即使最终化简有误，清晰地展示这些步骤也能获得部分分数。始终检查指数是正数、负数还是分数，并与原表达式的形式保持一致来书写中间结果。
11. Practice Questions | 练习题
Test your understanding with the following exam-style questions. Solutions are provided below for self-assessment.
请用以下考试风格的题目测试你的理解。答案提供在下方供自我评估。
Question 1: Differentiate y = 5x⁶ – 3x⁴ + 2x – 8.
题目1：求 y = 5x⁶ – 3x⁴ + 2x – 8 的导数。
Question 2: Differentiate y = 4/√x.
题目2：求 y = 4/√x 的导数。
Question 3: Differentiate y = (2x³ – 1)⁷.
题目3：求 y = (2x³ – 1)⁷ 的导数。
Question 4: Find the gradient of the tangent to the curve y = x³ – 6x² + 9x at x = 2.
题目4：求曲线 y = x³ – 6x² + 9x 在 x = 2 处切线的斜率。
Solutions: 1) dy/dx = 30x⁵ – 12x³ + 2. 2) Rewrite as y = 4x^(-1/2), dy/dx = -2x^(-3/2) = -2/(√x³) = -2/(x√x). 3) dy/dx = 7(2x³ – 1)⁶ × 6x² = 42x²(2x³ – 1)⁶. 4) dy/dx = 3x² – 12x + 9; at x = 2, dy/dx = 3(4) – 12(2) + 9 = 12 – 24 + 9 = -3.
答案：1) dy/dx = 30x⁵ – 12x³ + 2。2) 改写为 y = 4x^(-1/2)，dy/dx = -2x^(-3/2) = -2/(√x³) = -2/(x√x)。3) dy/dx = 7(2x³ – 1)⁶ × 6x² = 42x²(2x³ – 1)⁶。4) dy/dx = 3x² – 12x + 9；在 x = 2 处，dy/dx = 3(4) – 12(2) + 9 = 12 – 24 + 9 = -3。
12. Summary: The Complete Picture | 总结：完整图景
The power rule dy/dx = n·xⁿ⁻¹ is a compact formula with far-reaching applications. It works for every real exponent — positive, negative, fractional, and irrational. When combined with the sum rule, constant multiple rule, chain rule, and implicit differentiation, it becomes an indispensable tool for solving a wide range of differentiation problems.
幂函数法则 dy/dx = n·xⁿ⁻¹ 是一个简洁而应用广泛的公式。它对所有实数指数都适用——正数、负数、分数和无理数。当与加法法则、常数倍法则、链式法则和隐函数求导相结合时，它成为解决各类微分问题不可或缺的工具。
The key to mastery is recognising the underlying pattern: bring down the exponent as a multiplier, reduce the exponent by one, and simplify. Practice rewriting radicals and reciprocals as powers, and always check whether the chain rule is needed when the base is a function of x rather than x itself.
掌握的关键在于识别其内在模式：将指数降为乘数，指数减一，然后化简。练习将根号与倒数改写为幂的形式，并始终检查当底数是 x 的函数而非 x 本身时是否需要使用链式法则。
Mastering this single rule will serve you across all areas of A-Level Mathematics, from pure calculus to applications in physics and statistics. Keep practising, and the power rule will become second nature.
掌握这一条法则将在A-Level数学的各个领域为你提供帮助，从纯微积分到物理和统计学中的应用。持续练习，幂函数法则将成为你的第二天性。
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