📚 Differentiation from First Principles | 从第一原理求导
In A-Level Mathematics, differentiation is the cornerstone of calculus, enabling us to study rates of change, slopes of curves, and optimisation. Understanding the derivative from first principles gives a rigorous foundation, connecting the abstract limit concept to practical rules. This article thoroughly explores differentiation starting from the limit definition, derives basic functions, and extends to derivative rules with worked examples tailored to the Edexcel specification.
在A-Level数学中，微分是微积分的基石，使我们能够研究变化率、曲线斜率和最优化。从第一原理理解导数提供了一个严谨的基础，将抽象的极限概念与实用规则联系起来。本文从极限定义出发深入探讨微分，推导基本函数，并结合Edexcel考试规范，通过实例拓展到求导法则。
1. The Concept of Derivative and Slope | 导数与斜率的概念
presents the gradient of the tangent line to the graph y = f(x) at that point. For a straight line, gradient is constant, but for a curve, gradient varies and can be found by considering the limit of secant slopes.
函数在某一点的导数衡量函数关于其自变量的瞬时变化率。从几何角度看，它表示该点处曲线 y = f(x) 的切线斜率。对于直线，斜率是恒定的，但对于曲线，斜率是变化的，可以通过割线斜率的极限来求得。
If we take two points on the curve, (x, f(x)) and (x+h, f(x+h)), the gradient of the secant line joining them is (f(x+h) − f(x))/h. As h approaches 0, the secant becomes the tangent, and this limiting value is the derivative, denoted f'(x) or dy/dx.
如果我们在曲线上取两点 (x, f(x)) 和 (x+h, f(x+h))，连接它们的割线斜率为 (f(x+h) − f(x))/h。当 h 趋近于 0 时，割线变成切线，这个极限值就是导数，记作 f'(x) 或 dy/dx。
2. First Principles Definition | 第一原理定义
The formal definition of the derivative from first principles is:
从第一原理出发，导数的正式定义如下：
f'(x) = limₕ→₀ [f(x+h) − f(x)] / h
provided the limit exists. This definition requires evaluating the difference quotient and then taking the limit as h tends to zero. It applies to any differentiable function and forms the basis for proving standard differentiation rules.
前提是极限存在。该定义需要计算差商，然后取 h 趋于零时的极限。它适用于任何可微函数，并为证明标准求导法则奠定了基础。
Using this definition, we can derive the derivative of constant functions, power functions, trigonometric functions, and more. The process often involves algebraic manipulation to eliminate h in the denominator before applying the limit.
利用这个定义，我们可以推导出常数函数、幂函数、三角函数等的导数。这个过程通常涉及代数变形，以便在应用极限之前消去分母中的 h。
3. Derivative of a Constant | 常数的导数
Let f(x) = c, where c is a constant. Then f(x+h) = c. Substituting into the definition:
设 f(x) = c，其中 c 是常数。则 f(x+h) = c。代入定义：
f'(x) = limₕ→₀ (c − c)/h = limₕ→₀ 0/h = 0
Hence, the derivative of any constant is zero. Geometrically, a constant function is a horizontal line with zero gradient.
因此，任何常数的导数都是零。从几何上看，常数函数是一条水平线，斜率为零。
4. Differentiating xⁿ using First Principles | 利用第一原理求导 xⁿ
Consider f(x) = xⁿ for a positive integer n. The difference quotient becomes:
考虑正整数 n 的情形 f(x) = xⁿ。差商变为：
f'(x) = limₕ→₀ [(x+h)ⁿ − xⁿ] / h
Using the binomial expansion: (x+h)ⁿ = xⁿ + n xⁿ⁻¹h + [n(n-1)/2] xⁿ⁻²h² + ... + hⁿ. Then:
利用二项式展开：(x+h)ⁿ = xⁿ + n xⁿ⁻¹h + [n(n-1)/2] xⁿ⁻²h² + ... + hⁿ。于是：
(x+h)ⁿ − xⁿ = n xⁿ⁻¹h + [n(n-1)/2] xⁿ⁻²h² + ... + hⁿ
viding by h gives n xⁿ⁻¹ + terms containing h. As h → 0, all terms with h vanish, leaving:
除以 h 得到 n xⁿ⁻¹ + 含有 h 的项。当 h → 0 时，所有含 h 的项消失，剩下：
f'(x) = n xⁿ⁻¹
This proves the power rule for positive integers. The rule extends to all real exponents under appropriate conditions, a fact used freely at A-Level.
这证明了正整数情况下的幂函数求导法则。该法则在适当的条件下扩展到所有实数指数，在A-Level中可以自由使用。
5. Differentiating sin(x) from First Principles | 从第一原理求导 sin(x)
For f(x) = sin(x), use the first principles definition:
对于 f(x) = sin(x)，使用第一原理定义：
f'(x) = limₕ→₀ [sin(x+h) − sin(x)] / h
onometric identity sin(A) − sin(B) = 2 cos((A+B)/2) sin((A−B)/2). Here A = x+h, B = x, so:
应用三角恒等式 sin(A) − sin(B) = 2 cos((A+B)/2) sin((A−B)/2)。这里 A = x+h，B = x，所以：
sin(x+h) − sin(x) = 2 cos(x + h/2) sin(h/2)
Thus the derivative becomes:
因此导数变为：
f'(x) = limₕ→₀ [2 cos(x + h/2) sin(h/2)] / h = limₕ→₀ cos(x + h/2) · [sin(h/2) / (h/2)]
the standard limit limₜ→₀ sin(t)/t = 1 with t = h/2, and continuity of cosine, we obtain:
利用标准极限 limₜ→₀ sin(t)/t = 1，其中 t = h/2，以及余弦的连续性，得到：
f'(x) = cos(x) · 1 = cos(x)
). This derivation is a classic example of combining trigonometric identities with limits.
因此，d/dx[sin(x)] = cos(x)。这个推导是结合三角恒等式与极限的经典例子。
6. Differentiating cos(x) from First Principles | 从第一原理求导 cos(x)
for f(x) = cos(x), use the identity cos(A) − cos(B) = −2 sin((A+B)/2) sin((A−B)/2). Then:
类似地，对于 f(x) = cos(x)，使用恒等式 cos(A) − cos(B) = −2 sin((A+B)/2) sin((A−B)/2)。则有：
cos(x+h) − cos(x) = −2 sin(x + h/2) sin(h/2)
Therefore:
因此：
f'(x) = limₕ→₀ [−2 sin(x + h/2) sin(h/2)] / h = limₕ→₀ −sin(x + h/2) · [sin(h/2) / (h/2)]
using lim sin(t)/t = 1 and evaluating the limit yields −sin(x). So d/dx[cos(x)] = −sin(x).
再次利用 lim sin(t)/t = 1 并求极限得到 −sin(x)。所以 d/dx[cos(x)] = −sin(x)。
7. Differentiating eˣ and ln(x) from First Principles | 从第一原理求导 eˣ 和 ln(x)
For f(x) = eˣ, the first principles approach requires the property that the limit of (eʰ as h → 0 is 1. This special limit defines the natural exponential function. Substituting:
对于 f(x) = eˣ，第一原理方法需要性质：当 h → 0 时 (eʰ − 1)/h 的极限为 1。这个特殊极限定义了自然指数函数。代入：
f'(x) = limₕ→₀ [eˣ⁺ʰ − eˣ] / h = eˣ · limₕ→₀ (eʰ − 1)/h = eˣ · 1 = eˣ
Thus eˣ hip or first principles with a substitution leading to d/dx[ln(x)] = 1/x, valid for x > 0.
因此 eˣ 的导数等于其自身，这是一个独特的性质。对于 f(x) = ln(x)，可以利用反函数关系或通过第一原理进行代换，得到 d/dx[ln(x)] = 1/x，对 x > 0 成立。
8. Differentiating xⁿ for Rational Exponents | 有理指数 xⁿ 的求导
iation or first principles with binomial series for fractional powers. For example, f(x) = √x = x¹/². Using the definition:
幂函数法则可通过隐函数微分或利用二项式级数的第一原理推广到有理指数。例如，f(x) = √x = x¹/²。使用定义：
f'(x) = limₕ→₀ [√x+h − √x] / h
Rationalise the numerator: multiply numerator and denominator by √x+h + √x. This gives:
分子有理化：分子分母同乘 √x+h + √x。得到：
f'(x) = limₕ→₀ [(x+h) − x] / [h(√x+h + √x)] = limₕ→₀ 1 / (√x+h + √x) = 1/(2√x)
ⁿ⁻¹ with n = 1/2. Similar algebraic tricks confirm the power rule for all rational powers.
这与 n = 1/2 时的 n xⁿ⁻¹ 一致。类似的代数技巧可以确认所有有理次幂的幂函数法则。
9. Derivative Rules: Sum, Product, and Quotient | 导数法则：和、积与商
rst principles, we can prove differentiation rules. For sum: if h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x). The proof follows directly from the limit of a sum.
在建立了第一原理之后，我们可以证明微分法则。对于和：如果 h(x) = f(x) + g(x)，则 h'(x) = f'(x) + g'(x)。证明直接来自和的极限。
The product rule states: if y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v' el candidates are expected to use, not prove, but understanding the logic deepens insight.
乘积法则指出：如果 y = u(x)v(x)，则 dy/dx = u'(x)v(x) + u(x)v'(x)。可以通过考虑 f(x+h)g(x+h) − f(x)g(x) 并巧妙加上零项 f(x+h)g(x) − f(x+h)g(x) 来推导。Edexcel考生须会使用，而非证明，但理解逻辑能加深洞察。
The quotient rule: y = u/v ⇒ dy/dx = (u'v − uv')/v². Proved similarly using the limit definition and common denominator manipulation.
商法则：y = u/v ⇒ dy/dx = (u'v − uv')/v²。类似地使用极限定义和通分操作证明。
10. Chain Rule and Composite Functions | 链式法则与复合函数
The chain rule for y = f(g(x)) states dy/dx = f'(g(x)) · g' es careful handling of cases where Δu = 0, but the intuitive approach suffices at A-Level.
对于 y = f(g(x)) 的链式法则为 dy/dx = f'(g(x)) · g'(x)。用第一原理的语言，可以通过书写 Δy/Δx = (Δy/Δu)·(Δu/Δx)（其中 u = g(x)）并取极限来推导。严格证明需要仔细处理 Δu = 0 的情况，但在A-Level中直观方法已足够。
The chain rule is essential for differentiating functions like sin(3x), e²ˣ, or √x²+1. It is the most powerful tool for differentiating composite expressions encountered in the Edexcel syllabus.
链式法则对于微分诸如 sin(3x)、e²ˣ 或 √x²+1 等函数至关重要。它是处理Edexcel大纲中复合表达式最强大的工具。
11. Second Derivatives and Concavity | 二阶导数与凹凸性
Differentiating a derivative yields the second derivative, f''(x) or d²y/dx². It measures the rate of change of gradient and indicates concavity of the graph: if f''(x) > 0, the graph is convex (∪-shaped); if f''(x) < 0, it is concave (∩-shaped).
对导数求导得到二阶导数，记作 f''(x) 或 d²y/dx²。它衡量斜率的变化率，并指示图形的凹凸性：若 f''(x) > 0，图像上凸（∪形）；若 f''(x) < 0，图像下凹（∩形）。
Second derivatives are used to classify stationary points found by setting f' in kinematics: if displacement s = f(t), velocity v = ds/dt and acceleration a = d²s/dt².
二阶导数用于对令 f'(x)=0 求得的驻点进行分类。它们也出现在运动学中：若位移 s = f(t)，则速度 v = ds/dt，加速度 a = d²s/dt²。
12. Tangents, Normals, and Applications | 切线、法线及其应用
Once the derivative f'(x₀) is found, the equation of the tangent at (x₀, y₀) is y − y₀ = m (x − x₀) where m = f' al is perpendicular, so its gradient is −1/m, giving the equation y − y₀ = (−1/m)(x − x₀).
一旦求得导数 f'(x₀)，在点 (x₀, y₀) 处的切线方程为 y − y₀ = m (x − x₀)，其中 m = f'(x₀)。法线垂直于切线，因此斜率为 −1/m，方程为 y − y₀ = (−1/m)(x − x₀)。
Differentiation helps find maximum and minimum values of functions, essential for optimisation problems. For example, minimising surface area for a given volume or maximising profit. The link between first principles and these practical uses underscores the power of calculus.
微分有助于找到函数的最大值和最小值，这对最优化问题至关重要。例如，在给定体积下最小化表面积或最大化利润。第一原理与这些实际应用之间的联系凸显了微积分的力量。
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