Differentiation from First Principles | 从第一性原理求导

📚 Differentiation from First Principles | 从第一性原理求导

Differentiation from first principles is a core idea in Edexcel A Level Mathematics. It shows how the derivative of a function is defined by a limit of the gradient of a chord as two points on a curve become infinitely close. Understanding this definition helps you make sense of all the standard differentiation rules.

从第一性原理求导是爱德思 A Level 数学的核心思想。它展示了函数的导数如何由弦的斜率在曲线上两点无限接近时的极限来定义。理解这一定义有助于你理解所有标准求导法则。

1. The Derivative as a Limit | 导数作为极限

The derivative measures the instantaneous rate of change of a function at a point. On a graph, it is the gradient of the tangent to the curve at that point.

导数衡量函数在某一点的瞬时变化率。在图像上，它是曲线在该点处切线的斜率。

To find this tangent gradient, we first consider a chord joining two nearby points on the curve. As one point moves closer to the other, the chord gradient approaches the tangent gradient. This limiting process is called differentiation from first principles.

为了求出这条切线斜率，我们首先考虑连接曲线上两个邻近点的弦。随着一个点向另一个点靠近，弦的斜率会趋近于切线斜率。这个极限过程就叫做从第一性原理求导。

For example, if you plot y = x² and draw a chord between x = 1 and x = 1.1, the chord gradient is close to the tangent gradient at x = 1. As the second x-value moves to 1, the chord gradient becomes exactly 2, which is the tangent gradient.

例如，如果画出 y = x² 并在 x = 1 与 x = 1.1 之间作一条弦，弦的斜率会接近 x = 1 处切线的斜率。当第二个 x 值移动到 1 时，弦的斜率正好变成 2，这就是切线斜率。

gradient of chord = [f(x + h) − f(x)] ÷ h

2. The Formal Definition | 正式定义

Let f(x) be a function defined on an interval containing x. The derivative of f at x is given by the limit:

设 f(x) 是在包含 x 的区间上有定义的函数。f 在 x 处的导数由以下极限给出：

f ‘(x) = lim(h → 0) [f(x + h) − f(x)] ÷ h

Here h represents a small change in x. The expression f(x + h) − f(x) is the change in y, so the fraction is the gradient of the chord between the points (x, f(x)) and (x + h, f(x + h)).

这里 h 表示 x 的一个小增量。表达式 f(x + h) − f(x) 是 y 的变化量，因此该分式就是点 (x, f(x)) 与点 (x + h, f(x + h)) 之间弦的斜率。

When h tends to 0, the chord becomes the tangent, provided the limit exists.

当 h 趋于 0 时，弦就变成切线，前提是该极限存在。

If the limit exists, we say f is differentiable at x. If the left-hand limit and right-hand limit are different, the derivative does not exist at that point.

如果该极限存在，我们称 f 在 x 处可导。如果左极限和右极限不同，那么该点的导数不存在。

3. Step-by-Step Method | 分步方法

The standard method for differentiating from first principles can be broken into four clear steps.

从第一性原理求导的标准方法可分为四个清晰的步骤。

• Write down f(x + h), then find f(x + h) − f(x). / 写出 f(x + h)，然后求出 f(x + h) − f(x)。
• Divide the result by h and simplify the expression. / 将结果除以 h 并化简表达式。
• Take the limit as h approaches 0. / 令 h 趋于 0 取极限。
• State the derivative f ‘(x). / 写出导数 f ‘(x)。

Each example in the next sections follows this exact structure, so you can use it as a template in the exam.

接下来几节中的每个例子都遵循这一结构，因此你可以将其作为考试中的模板。

Always simplify the fraction before substituting h = 0, because direct substitution often gives the indeterminate form 0/0.

在代入 h = 0 之前一定要化简分式，因为直接代入通常会得到不确定形式 0/0。

4. Example: f(x) = x² | 示例：f(x) = x²

For f(x) = x², we write f(x + h) = (x + h)² = x² + 2xh + h².

对于 f(x) = x²，我们写出 f

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