📚 Differentiation from First Principles | 导数定义法求导
In this article, we explore the foundational method of differentiation: using the formal limit definition of a derivative. This approach, often called “differentiation from first principles,” reveals the true meaning of the derivative and provides a rigorous basis for all differentiation rules.
本文将深入探讨求导的基础方法——利用导数的形式化极限定义。这种方法通常被称为“导数定义法”或“第一性原理求导”，它揭示了导数的本质，并为所有求导法则提供了严谨的理论基础。
1. The Definition of the Derivative | 导数的定义
The derivative of a function \( f(x) \) at a point \( x \) is defined as the limit of the difference quotient as \( h \) approaches zero:
函数 \( f(x) \) 在点 \( x \) 处的导数定义为差商的极限（当 \( h \) 趋于零时）：
f'(x) = limh→0 [f(x+h) − f(x)] / h
This limit, if it exists, gives the instantaneous rate of change of the function with respect to \( x \). We say that \( f \) is differentiable at \( x \) when this limit exists.
如果该极限存在，它便给出了函数关于 \( x \) 的瞬时变化率。我们称 \( f \) 在 \( x \) 处可导（或可微），当且仅当这个极限存在。
2. Geometric Interpretation | 几何意义
Geometrically, the difference quotient \( [f(x+h) − f(x)] / h \) represents the slope of the secant line through the two points \( (x, f(x)) \) and \( (x+h, f(x+h)) \). As \( h \) approaches zero, this secant line approaches the tangent line at \( x \), and its slope approaches the derivative.
从几何上看，差商 \( [f(x+h) − f(x)] / h \) 表示经过两点 \( (x, f(x)) \) 和 \( (x+h, f(x+h)) \) 的割线斜率。当 \( h \) 趋近于零时，这条割线趋近于点 \( x \) 处的切线，其斜率也趋近于导数值。
The derivative \( f'(x) \) is therefore the slope of the tangent line to the curve \( y = f(x) \) at the point \( (x, f(x)) \).
因此，导数 \( f'(x) \) 就是曲线 \( y = f(x) \) 在点 \( (x, f(x)) \) 处切线的斜率。
3. Why Use First Principles? | 为什么使用定义法？
First principles is not merely an exercise; it is the origin of every differentiation rule. Understanding this definition allows you to derive rules such as the power rule, product rule, and chain rule from scratch, rather than memorising them without proof.
定义法不仅仅是一种练习，它是所有求导法则的源头。理解这一定义，你就能从零推导出幂法则、乘积法则、链式法则等规则，而不是死记硬背而不知其所以然。
Moreover, the definition helps you determine whether a function is differentiable at a given point, especially when the function involves absolute values, piecewise definitions, or sharp corners.
此外，定义法还能帮助你判断函数在某一点是否可导，特别是当函数包含绝对值、分段定义或尖点等情况时。
4. Derivative of a Constant Function | 常数函数的导数
Let \( f(x) = c \), where \( c \) is a constant. Then for any \( h eq 0 \):
设 \( f(x) = c \)，其中 \( c \) 为常数。则对任意 \( h ≠ 0 \)：
f'(x) = limh→0 [c − c] / h = limh→0 0 / h = 0
Thus the derivative of a constant is always zero. This matches the intuitive idea that a constant function has no change.
因此常数的导数恒为零。这与常数函数没有变化这一直观概念一致。
5. Derivative of a Linear Function | 线性函数的导数
Let \( f(x) = ax + b \). Compute the difference quotient:
设 \( f(x) = ax + b \)。计算差商：
[a(x+h) + b − (ax + b)] / h = [ah] / h = a
Taking the limit as \( h → 0 \) gives \( f'(x) = a \). The slope of a straight line is constant, so its derivative is simply the slope.
令 \( h → 0 \) 取极限，得 \( f'(x) = a \)。直线的斜率是恒定的，因此其导数就是斜率本身。
6. Derivative of \( x^2 \) | \( x^2 \) 的导数
Let \( f(x) = x^2 \). Then:
设 \( f(x) = x^2 \)。则：
(x+h)² − x² = x² + 2xh + h² − x² = 2xh + h²
Divide by \( h \) to get \( 2x + h \), and take the limit \( h → 0 \):
除以 \( h \) 得 \( 2x + h \)，再令 \( h → 0 \) 取极限：
f'(x) = limh→0 (2x + h) = 2x
This is the simplest example of the power rule: for \( x^n \), the derivative is \( nx^{n−1} \).
这是幂法则最简单的例子：对于 \( x^n \)，其导数为 \( nx^{n−1} \)。
7. Derivative of \( x^n \) (Positive Integer n) | \( x^n \) 的导数（n 为正整数）
Using the binomial expansion, for a positive integer \( n \):
利用二项式展开，对于正整数 \( n \)：
(x+h)ⁿ = xⁿ + n xⁿ⁻¹ h + [n(n−1)/2] xⁿ⁻² h² + … + hⁿ
Then the difference quotient becomes:
于是差商变为：
[(x+h)ⁿ − xⁿ] / h = n xⁿ⁻¹ + [n(n−1)/2] xⁿ⁻² h + … + hⁿ⁻¹
As \( h → 0 \), all terms containing \( h \) vanish, leaving:
当 \( h → 0 \) 时，所有含 \( h \) 的项均趋于零，剩下：
f'(x) = n xⁿ⁻¹
This derivation works for any real exponent \( n \) as well, although the binomial expansion must be replaced by the generalised binomial theorem.
这一推导对任意实数指数 \( n \) 也成立，只是需要将二项式展开推广为广义二项式定理。
8. Derivative of \( 1/x \) | \( 1/x \) 的导数
Let \( f(x) = 1/x \), with \( x eq 0 \). Then:
设 \( f(x) = 1/x \)，其中 \( x ≠ 0 \)。则：
[1/(x+h) − 1/x] / h = [(x − (x+h)) / (x(x+h))] / h = [−h / (x(x+h))] / h = −1 / [x(x+h)]
Taking the limit as \( h → 0 \):
令 \( h → 0 \) 取极限：
f'(x) = −1 / x²
This matches the power rule: \( x^{-1} \) has derivative \( −x^{-2} \).
这与幂法则一致：\( x^{-1} \) 的导数为 \( −x^{-2} \)。
9. Derivative of \( \sqrt{x} \) | \( \sqrt{x} \) 的导数
Let \( f(x) = √x \), with \( x > 0 \). Rationalise the numerator:
设 \( f(x) = √x \)，其中 \( x > 0 \)。对分子进行有理化：
[√(x+h) − √x] / h = [(√(x+h) − √x)(√(x+h) + √x)] / [h (√(x+h) + √x)]
= h / [h (√(x+h) + √x)] = 1 / [√(x+h) + √x]
As \( h → 0 \), the denominator approaches \( 2√x \), so:
当 \( h → 0 \) 时，分母趋于 \( 2√x \)，因此：
f'(x) = 1 / (2√x)
Notice that \( \sqrt{x} = x^{1/2} \), and the power rule gives \( \frac{1}{2} x^{-1/2} \), which is identical.
注意 \( \sqrt{x} = x^{1/2} \)，幂法则给出 \( \frac{1}{2} x^{-1/2} \)，与之完全相同。
10. Differentiability and Continuity | 可导性与连续性
A function must be continuous at a point to be differentiable there, but continuity alone is not sufficient. For example, \( f(x) = |x| \) is continuous at \( x = 0 \), yet the left and right limits of the difference quotient are \( −1 \) and \( 1 \), so the derivative does not exist.
函数在某点可导，则必须先在该点连续，但连续并不足以保证可导。例如 \( f(x) = |x| \) 在 \( x = 0 \) 处连续，但差商的左、右极限分别为 \( −1 \) 和 \( 1 \)，因此该点导数不存在。
To test differentiability at a point, check whether the two-sided limit of the difference quotient exists:
要检验某点的可导性，需检查差商的双侧极限是否存在：
limh→0⁻ [f(x+h) − f(x)] / h = limh→0⁺ [f(x+h) − f(x)] / h
If both one-sided limits are equal, the derivative exists; otherwise it does not.
若左右极限相等，则导数存在；否则导数不存在。
11. Common Mistakes and Tips | 常见错误与提示
When using first principles, students often make algebra errors. Here are key points to remember:
使用定义法时，学生常犯代数错误。以下是要点提醒：
- Write the limit symbol at every step until you actually substitute \( h = 0 \). Do not drop it prematurely.
- 在每一步保留极限符号，直到真正代入 \( h = 0 \) 为止，不要过早省略。
- Simplify the difference quotient completely before taking the limit. If direct substitution gives \( 0/0 \), factorise or rationalise.
- 在取极限前，先彻底化简差商。若直接代入得到 \( 0/0 \)，应进行因式分解或有理化。
- For expressions with \( h \) in the denominator, never set \( h = 0 \) before simplifying – that would be undefined.
- 当分母中含有 \( h \) 时，切勿在化简前令 \( h = 0 \)——那将导致未定义。
- Use the correct expansion for \( (x+h)^n \). For \( n = 2, 3 \), expand explicitly; for general \( n \), use the binomial theorem.
- 正确展开 \( (x+h)^n \)。当 \( n = 2, 3 \) 时逐项展开；对一般 \( n \)，使用二项式定理。
12. Summary | 总结
Differentiation from first principles is the rigorous definition of the derivative as a limit. It connects the geometric idea of a tangent line to the algebraic concept of instantaneous rate of change. Mastering this method strengthens your understanding of calculus and prepares you for more advanced topics such as limits, continuity, and higher-order derivatives.
导数定义法将导数定义为极限，是严格而本质的求导方法。它将切线的几何直观与瞬时变化率的代数概念紧密联系。掌握这一方法能加深你对微积分的理解，并为学习极限、连续性、高阶导数等进阶内容奠定坚实基础。
Practice by deriving the derivative of many functions using first principles, and you will gain both computational skill and conceptual insight.
请通过定义法亲自推导多种函数的导数，在练习中同时提升计算技巧与概念理解。
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