Differentiating x^n | 微分 xⁿ

📚 Differentiating xⁿ | 微分 xⁿ

Differentiation is a cornerstone of calculus, allowing us to understand how a function changes at any instant. For A-Level Mathematics, mastering the derivative of xⁿ forms the foundation for tackling polynomials, rational functions, and even more advanced expressions. This article explores the rule from first principles, covers fractional and negative powers, and demonstrates key applications throughout the Edexcel syllabus.

微分是微积分的基石，让我们得以理解函数在任意瞬时的变化。在 A-Level 数学中，掌握 xⁿ 的导数构成了处理多项式、有理函数乃至更复杂表达式的基础。本文将从第一原理出发推导该法则，涵盖分数次幂与负次幂，并演示其在 Edexcel 大纲中的关键应用。

1. Introduction to Differentiation | 微分简介

aneous rate of change of a quantity. For a function y = f(x), the derivative, denoted by f'(x) or dy/dx, gives the gradient of the tangent line at any point along the curve.

微分衡量一个量的瞬时变化率。对于函数 y = f(x)，导数记作 f′(x) 或 dy/dx，它给出了曲线上任意一点处切线的斜率。

The concept arises naturally from the idea of a limit: we consider the average rate of change over a tiny interval and then let that interval shrink to zero. This process leads to the derivative function, which itself can be differentiated further to obtain higher-order derivatives.

这一概念源自极限思想：我们考察一个极小间隔内的平均变化率，再让该间隔趋于零。这一过程便产生了导函数，而导函数本身还可继续微分以得到高阶导数。

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2. The Limit Definition of Derivative | 导数的极限定义

The formal definition of the derivative of f(x) is given by the limit:

导数的正式定义为下列极限：

f′(x) = limₕ→₀ (f(x+h) − f(x)) / h

provided this limit exists. This expression captures the slope of the secant line through two points on the curve as those points become infinitely close.

只要该极限存在。此表达式捕捉了曲线上两点间割线的斜率，当这两点无限接近时的极限情形。

tly and expand using the binomial theorem. This rigorous approach confirms the simple power rule we will use in practice.

对于 f(x) = xⁿ，可直接代入并利用二项式定理展开。这种严密的方法验证了我们将使用的简洁幂法则。

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3. Deriving the Derivative of xⁿ from First Principles | 从第一原理推导 xⁿ 的导数

Take f(x) = xⁿ. Then f(x+h) = (x+h)ⁿ. Expanding via the binomial theorem gives:

取 f(x) = xⁿ，则 f(x+h) = (x+h)ⁿ。通过二项式定理展开可得：

(x+h)ⁿ = xⁿ + n xⁿ⁻¹ h + (ⁿC₂) xⁿ⁻² h² + … + hⁿ

n xⁿ⁻¹ h + higher-order terms in h. Dividing by h gives n xⁿ⁻¹ + (terms containing h, h², …). As h → 0, every term containing h vanishes, leaving precisely n xⁿ⁻¹.

减去 f(x)=xⁿ 后得到 f(x+h)−f(x) = n xⁿ⁻¹ h + 含 h 的更高阶项。除以 h 得到 n xⁿ⁻¹ +（含 h, h², … 的项）。当 h → 0 时，所有含 h 的项都消失，恰好剩下 n xⁿ⁻¹。

Hence, the derivative is established without any hand-waving. This proof works for positive integer n, but the result generalises to all real n.

因此，导数被毫不含混地推导出来。该证明适用于正整数 n，但结果可推广至所有实数 n。

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4. The Power Rule | 幂法则

The foundational rule for differentiating powers of x is:

微分 x 的幂次的基本法则为：

d/dx (xⁿ) = n xⁿ⁻¹

where n can be any real number constant. This is often called the "power rule" and is the first derivative rule students learn.

其中 n 可为任意实数常数。这通常被称为“幂法则”，是学生最先学习的求导法则。

(x) = 1·x⁰ = 1, which aligns with the fact that the line y = x has a constant slope of 1.

当 n = 1 时，该法则给出 d/dx (x) = 1·x⁰ = 1，这与直线 y = x 的恒定斜率 1 相符。

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5. Differentiating Positive Integer Powers | 微分正整数次幂

For small integer powers, the rule operates simply:

对于较小的整数次幂，该法则操作简单：

• d/dx (x²) = 2x
• d/dx (x³) = 3x²
• d/dx (x⁵) = 5x⁴

• d/dx (x²) = 2x
• d/dx (x³) = 3x²
• d/dx (x⁵) = 5x⁴

Each time, the old power multiplies the term and the new power is one less. This pattern makes polynomial differentiation highly systematic.

每一次，原来的指数乘到前面，而新指数减一。这种模式使多项式的微分变得非常系统化。

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6. Fractional Powers (Roots) | 分数次幂（根式）

The power rule extends seamlessly to fractional exponents, which represent roots. Using the notation x½ for √x and x⅓ for ∛x, we obtain:

幂法则可无缝扩展到分数指数，这些指数表示根式。用 x½ 表示 √x，用 x⅓ 表示 ∛x，我们得到：

d/dx (x½) = ½ x⁻½ = 1 / (2√x)

d/dx (x⅓) = ⅓ x⁻⅔ = 1 / (3 x²⁄₃)

This approach avoids having to remember separate rules for square roots or cube roots. For example, the derivative of √x is 1/(2√x).

这种方法避免了为平方根或立方根记忆单独的规则。例如，√x 的导数为 1/(2√x)。

More generally, d/dx (x^p/q) = (p/q) x^{p/q − 1}. As long as the exponent is a rational number, the power rule holds, making it an indispensable tool.

更一般地，d/dx (x^p/q) = (p/q) x^{p/q − 1}。只要指数是有理数，幂法则就成立，使之成为不可或缺的工具。

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7. Negative Powers (Reciprocals) | 负次幂（倒数）

If n is negative, the rule gives derivatives of expressions like 1/x or 1/x². Rewriting 1/x as x⁻¹, we have:

若 n 为负值，该法则给出诸如 1/x 或 1/x² 等表达式的导数。将 1/x 写作 x⁻¹，我们有：

d/dx (x⁻¹) = −1 x⁻² = −1 / x²

d/dx (x⁻²) = −2 x⁻³ = −2 / x³

Thus, the derivative of 1/x² becomes −2/x³. The key is always to lower the power by one, even when it is negative.

于是，1/x² 的导数变为 −2/x³。关键在于总是指数减一，即使指数为负。

such as n = −½, giving d/dx (x⁻½) = −½ x⁻³⁄₂, which finds use in physics and engineering.

这同样适用于更复杂的负指数，例如 n = −½，给出 d/dx (x⁻½) = −½ x⁻³⁄₂，在物理和工程中均有应用。

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8. Constant Multiples and Sums | 常数倍与和差法则

f c is a constant, then d/dx (c·f(x)) = c·f′(x). Also, d/dx (f(x) ± g(x)) = f′(x) ± g′(x).

微分算子是线性的；这意味着我们可以逐项微分。若 c 为常数，则 d/dx (c·f(x)) = c·f′(x)。此外，d/dx (f(x) ± g(x)) = f′(x) ± g′(x)。

differentiate any polynomial effortlessly. For instance, d/dx (4x³ − 2x² + 5) = 12x² − 4x.

结合幂法则，我们便能毫不费力地微分任意多项式。例如，d/dx (4x³ − 2x² + 5) = 12x² − 4x。

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9. Second Derivative and Higher Orders | 二阶导数及更高阶

The second derivative, written f''(x) or d²y/dx², is obtained by differentiating the first derivative. For xⁿ, applying the power rule twice gives:

二阶导数，记作 f′′(x) 或 d²y/dx²，由一阶导数再次微分得到。对于 xⁿ，两次应用幂法则得到：

f′′(x) = n(n−1) xⁿ⁻²

ourth, or kth derivatives. The pattern emerges clearly: the kth derivative of xⁿ is n(n−1)…(n−k+1) xⁿ⁻ᵏ, provided k ≤ n.

此过程可继续求得三阶、四阶或 k 阶导数。模式清晰呈现：xⁿ 的 k 阶导数为 n(n−1)…(n−k+1) xⁿ⁻ᵏ，其中 k ≤ n。

If k exceeds n, the derivative eventually becomes zero for positive integer n, because one factor in the product will be zero.

若 k 超过 n，对于正整数 n，由于乘积中某因子为零，导数最终将变为零。

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10. Tangents, Normals and Stationary Points | 切线、法线与驻点

The derivative f' ent of the tangent at x = a. The equation of the tangent line at (a, f(a)) is y − f(a) = f'(a)(x − a).

导数 f'(a) 给出了 x = a 处切线的斜率。在点 (a, f(a)) 处的切线方程为 y − f(a) = f'(a)(x − a)。

The normal line is perpendicular to the tangent; its gradient is −1/f'(a), assuming f'(a) ≠ 0. Setting f' points, which may be maxima, minima or points of inflection. The second derivative test f''(a) > 0 indicates a local minimum, while f''(a) < 0 indicates a local maximum.

法线垂直于切线；其斜率为 −1/f'(a)，假定 f'(a) ≠ 0。令 f'(x) = 0 有助于定位驻点，它们可能是极大值、极小值或拐点。二阶导数检验：f''(a) > 0 表明局部极小值，而 f''(a) < 0 表明局部极大值。

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11. Common Mistakes and How to Avoid Them | 常见错误及如何避免

Many students forget to multiply by the original power or mishandle the index arithmetic. For instance, an error is writing d/dx (x³) as 3x³ instead of 3x².

许多学生会忘记乘上原来的指数，或搞错指数运算。例如，将 d/dx (x³) 误写成 3x³ 而非 3x²。

the derivative of a constant is zero. Also, when dealing with fractional powers, students may forget to convert root notation correctly: always rewrite √x as x½ before differentiating.

另一个常见陷阱是忽略常数项：d/dx (xⁿ + c) = n xⁿ⁻¹，因为常数的导数为零。处理分数次幂时，学生也可能忘记正确转换根号记法：在微分前，务必将 √x 重新表示为 x½。

For sums, each term must be differentiated separately; trying to differentiate inside a bracket without expanding or using the chain rule often leads to mistakes.

对于和式，每一项必须分别微分；不展开括号就直接微分内部，或误用链式法则，常导致错误。

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12. Summary and Key Formulae | 总结与核心公式

Rule	Example	Derivative
Power Rule	xⁿ	n xⁿ⁻¹
Constant Multiple	c f(x)	c f′(x)
Sum/Difference	f(x) ± g(x)	f′(x) ± g′(x)
Constant	k	0
Second Derivative	f′(x) again	f′′(x)=n(n−1)xⁿ⁻²

Mastery of these essential rules allows you to differentiate any polynomial and a wide range of rational and root functions. Consistent practice linking the algebraic manipulation with the geometric interpretation of gradient, tangents and normals will build deep understanding for the A-Level exam.

掌握这些基本法则后，你就能微分任何多项式以及大量有理函数和根式函数。持续将代数操作与斜率、切线、法线等几何含义结合练习，将为 A-Level 考试培养深刻的理解。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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