Investigation 1 – Simple Rules of Differentiation | 探究1 – 简单的微分法则

📚 Investigation 1 – Simple Rules of Differentiation | 探究1 – 简单的微分法则

In this investigation, we explore the fundamental rules that govern the differentiation of basic functions. These simple rules—the constant rule, power rule, constant multiple rule, and sum/difference rule—form the backbone of differential calculus. Mastering them allows you to differentiate any polynomial and many other elementary functions quickly, without repeatedly resorting to the limit definition. This article will guide you through each rule, showing how they arise from first principles and how to apply them confidently in IB-style problems.

在本探究中，我们将探讨支配基本函数微分的基本法则。这些简单的法则——常数法则、幂法则、常数倍法则以及和差法则——构成了微分学的支柱。掌握它们能让你快速地对任何多项式及许多其他初等函数求导，而无需反复借助极限定义。本文将引导你逐一理解这些法则，展示它们如何从第一原理推导而来，并教你如何自信地在IB风格的问题中应用它们。

1. Understanding the Derivative | 理解导数

The derivative of a function f(x) at a point x measures the instantaneous rate of change, or the slope of the tangent line at that point. Formally, f'(x) = lim_h→0 (f(x+h) – f(x))/h, provided this limit exists. This definition, often called differentiation from first principles, is the foundation upon which all differentiation rules are built.

函数 f(x) 在点 x 处的导数衡量的是该点的瞬时变化率，或者切线的斜率。形式上，f'(x) = lim_h→0 (f(x+h) – f(x))/h，只要该极限存在。这一定义通常被称为从第一原理求导，是所有微分法则建立的基础。

For a simple power function like f(x) = x², we can compute the derivative using the limit: f'(x) = lim_h→0 ((x+h)² – x²)/h = lim_h→0 (2xh + h²)/h = lim_h→0 (2x + h) = 2x. This process shows the pattern that leads to the power rule.

对于一个简单的幂函数如 f(x) = x²，我们可以使用极限来计算导数：f'(x) = lim_h→0 ((x+h)² – x²)/h = lim_h→0 (2xh + h²)/h = lim_h→0 (2x + h) = 2x。这一过程展示了导致幂法则的模式。

However, repeating this limit for every function would be tedious. Instead, we derive general rules that make differentiation a mechanical process. This investigation focuses on those rules for simple functions.

然而，对每一个函数重复这一极限过程将是繁琐的。相反，我们推导出一般法则，使得微分成为一个机械化的过程。本次探究的重点正是针对简单函数的这些法则。

2. The Constant Rule | 常数法则

If f(x) = c, where c is any real number constant, then f'(x) = 0. Graphically, the function represents a horizontal line, so its slope is everywhere zero. Using the limit definition: f'(x) = lim_h→0 (c – c)/h = lim_h→0 0/h = 0.

如果 f(x) = c，其中 c 为任意实常数，则 f'(x) = 0。从图形上看，该函数代表一条水平线，因此其斜率处处为零。使用极限定义：f'(x) = lim_h→0 (c – c)/h = lim_h→0 0/h = 0。

This rule tells us that the derivative of any constant is zero. For example, d/dx (5) = 0, and d/dx (-π) = 0. It may seem trivial, but it is essential when differentiating polynomials containing constant terms.

该法则告诉我们，任何常数的导数均为零。例如，d/dx (5) = 0，以及 d/dx (-π) = 0。这看起来微不足道，但在对含有常数项的多项式求导时却是必不可少的。

3. The Power Rule | 幂法则

For a function of the form f(x) = xⁿ, where n is any real number, the derivative is f'(x) = n·xⁿ⁻¹. This rule is derived by extending the pattern observed in f(x) = x² and can be proven via the binomial theorem for positive integer n. For now, we accept it as a reliable shortcut.

对于形如 f(x) = xⁿ 的函数，其中 n 为任意实数，其导数为 f'(x) = n·xⁿ⁻¹。该法则通过扩展在 f(x) = x² 中观察到的模式得出，并可通过二项式定理对正整数 n 进行证明。目前我们将其作为一个可靠的便捷规则来接受。

Examples: If f(x) = x³, then f'(x) = 3x². If g(x) = x⁵, then g'(x) = 5x⁴. For a root function like h(x) = √x = x^(½), the power rule gives h'(x) = ½ x^(⁻½) = 1/(2√x). The rule also works for negative exponents: d/dx (x⁻²) = -2x⁻³.

示例：若 f(x) = x³，则 f'(x) = 3x²。若 g(x) = x⁵，则 g'(x) = 5x⁴。对于根式函数如 h(x) = √x = x^(½)，幂法则给出 h'(x) = ½ x^(⁻½) = 1/(2√x)。该法则同样适用于负指数：d/dx (x⁻²) = -2x⁻³。

The power rule is the most frequently used differentiation tool. It is important to remember to decrease the exponent by 1 and multiply by the original exponent.

幂法则是使用最频繁的微分工具。重要的是要记住，将指数减1并乘以原来的指数。

4. The Constant Multiple Rule | 常数倍法则

If a function is multiplied by a constant coefficient, the derivative is the constant times the derivative of the function. Formally, d/dx [c·f(x)] = c·f'(x), where c is a constant. This rule allows constants to pass through the differentiation operator unchanged.

如果一个函数乘以一个常数系数，那么导数就等于该常数乘以该函数的导数。正式地，d/dx [c·f(x)] = c·f'(x)，其中 c 为常数。该法则允许常数不变地通过微分运算符。

Using the limit definition: lim_h→0 (c·f(x+h) – c·f(x))/h = c·lim_h→0 (f(x+h)-f(x))/h = c·f'(x). This makes sense because scaling a function vertically just scales the slope by the same factor.

使用极限定义：lim_h→0 (c·f(x+h) – c·f(x))/h = c·lim_h→0 (f(x+h)-f(x))/h = c·f'(x)。这是合理的，因为将函数垂直缩放只是将斜率按相同比例缩放。

Example: d/dx (3x⁴) = 3·d/dx (x⁴) = 3·4x³ = 12x³. Another: d/dx (-5x²) = -5·2x = -10x. This rule is often combined with the power rule to differentiate monomials like axⁿ.

示例：d/dx (3x⁴) = 3·d/dx (x⁴) = 3·4x³ = 12x³。另一个：d/dx (-5x²) = -5·2x = -10x。该法则常与幂法则结合，用于对 axⁿ 这样的单项式求导。

5. The Sum and Difference Rules | 和差法则

The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives. That is, d/dx [f(x) ± g(x)] = f'(x) ± g'(x). This rule states that differentiation is a linear operation, allowing us to differentiate term-by-term.

函数和（或差）的导数等于它们各自导数的和（或差）。即 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。该法则表明，微分是一种线性运算，允许我们逐项求导。

Proof sketch: Using the limit definition and limit properties, the limit of a sum equals the sum of limits, provided each limit exists. Thus, it follows directly.

证明概述：利用极限定义和极限的性质，和的极限等于极限的和，只要每个极限存在。因此可以直接得出。

Example: d/dx (x³ + 2x²) = 3x² + 4x. Another: d/dx (4x

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