Differentiating x^n | 对 x^n 求导

📚 Differentiating x^n | 对 x^n 求导

Differentiation is a core tool in A-Level Mathematics. For Edexcel, one of the first rules you learn is how to differentiate powers of x. This rule appears in almost every calculus question, so mastering it is essential.

微积分是 A-Level 数学的核心工具。在 Edexcel 考试中，你最先学会的规则之一就是如何对 x 的幂求导。这条规则几乎出现在所有微积分题目中，因此掌握它至关重要。

1. The Definition of a Derivative | 导数的定义

The derivative f'(x) measures the rate at which f(x) changes with x. Geometrically, it gives the gradient of the tangent to the curve at any point.

导数 f'(x) 衡量 f(x) 随 x 变化的速率。从几何上看，它给出曲线上任意一点处切线的斜率。

By definition, the derivative is the limit of the gradient of a chord as the change in x tends to zero.

根据定义，导数是当 x 的改变量趋于零时，割线斜率的极限。

f'(x) = limₕ→₀ [f(x+h) − f(x)] / h

This definition is sometimes called ‘differentiation from first principles’. For f(x)=xⁿ, this becomes the starting point for proving the power rule.

这个定义有时被称为“用基础原理求导”。对于 f(x)=xⁿ，这也是证明幂函数求导法则的出发点。

2. The Power Rule | 幂函数求导法则

The power rule is the most important shortcut for differentiating powers of x. It states: if y = xⁿ, then the derivative is found by bringing the power down and then reducing the power by one.

幂函数求导法则是求 x 的幂导数的最重要捷径。它指出：若 y = xⁿ，则求导时将指数“拉下来”，然后把指数减一。

If y = xⁿ, then dy/dx = n xⁿ⁻¹

For example, the derivative of x³ is 3x², and the derivative of x⁵ is 5x⁴. The table below shows some common cases.

例如，x³ 的导数是 3x²，x⁵ 的导数是 5x⁴。下表给出了一些常见情形。

3. Proof for Positive Integer Powers | 正整数指数的证明

Let n be a positive integer. We can prove the power rule using the definition of the derivative and the binomial expansion.

设 n 为正整数。我们可以利用导数的定义和二项式展开来证明幂函数求导法则。

(x+h)ⁿ = xⁿ + n xⁿ⁻¹h + [n(n−1)/2]xⁿ⁻²h² + … + hⁿ

Substitute this into the first-principles formula and subtract xⁿ from the expansion. Then divide the result by h.

将这个展开式代入基础求导公式，并从展开式中减去 xⁿ，然后除以 h。

Every term after n xⁿ⁻¹h contains at least one factor h. As h tends to zero, all those terms vanish, leaving only n xⁿ⁻¹.

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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