📚 Differentiation from First Principles and Applications | 从第一原理出发的微分及其应用
Differentiation is one of the two central ideas in calculus, alongside integration. In AQA A-Level Mathematics, you are expected not only to differentiate standard functions confidently, but also to understand the limit definition from first principles, interpret the derivative as a gradient, and apply differentiation to tangents, normals, stationary points and real-world optimisation.
微分是微积分的两大核心思想之一，与积分并列。在 AQA A-Level 数学中，你不仅要熟练掌握标准函数的求导，还需要理解第一原理的极限定义、将导数解释为梯度，并把微分应用于切线、法线、驻点和现实情境中的最优化问题。
1. What is Differentiation? | 什么是微分？
gives the instantaneous rate of change of y with respect to x. Geometrically, it is the gradient of the tangent to the curve at a given point.
微分是用来求一个量相对于另一个量的变化率的过程。对于函数 y = f(x)，导数 dy/dx 表示 y 关于 x 的瞬时变化率。从几何角度看，它就是曲线在某点处切线的梯度。
2. Gradient of a Curve at a Point | 曲线上某点的梯度
A straight line has constant gradient, but a curve's gradient changes point to point. To estimate the gradient at point P, we take a nearby point Q and compute the slope of the chord PQ. As Q moves closer to P, the chord slope approaches the tangent slope. This limiting process is the foundation of differentiation.
直线有恒定梯度，但曲线的梯度逐点变化。为估计点 P 处的梯度，我们取附近的点 Q 并计算弦 PQ 的斜率。当 Q 逐渐靠近 P 时，弦的斜率趋近于切线的斜率。这一极限过程正是微分的基础。
Gradient of chord PQ = [f(x + h) − f(x)] / h
3. First Principles Formula | 第一原理公式
The derivative of f(x) from first principles is defined by the following limit:
函数 f(x) 从第一原理出发的导数由以下极限定义：
f′(x) = limₕ→₀ [f(x + h) − f(x)] / h
Here h is a small change in x. We substitute f(x + h) and f(x), simplify the difference quotient, and then let h tend to 0. The result is the derivative f′(x).
其中 h 是 x 的一个微小增量。我们代入 f(x + h) 和 f(x)，化简差商，然后令 h 趋于 0。所得结果即为导数 f′(x)。
+ h², so the difference quotient is (2xh + h²)/h = 2x + h. Letting h → 0 gives f′(x) = 2x.
例如，若 f(x) = x²，则 f(x + h) = (x + h)² = x² + 2xh + h²，因此差商为 (2xh + h²)/h = 2x + h。令 h → 0 得 f′(x) = 2x。
4. Differentiation Notation | 微分符号
Three common notations are used. f′(x), read as 'f dash of x', is called Lagrange notation. dy/dx, read as 'dy by dx', is Leibniz notation. d/dx(f(x)) is operator notation. In AQA exams, you need to be fluent in all three, especially when using the chain rule or differentiating implicitly.
常用的导数符号有三种。f′(x) 读作 'f dash of x'，称为拉格朗日符号。dy/dx 读作 'dy by dx'，是莱布尼茨符号。d/dx(f(x)) 是算子符号。在 AQA 考试中，你需要熟练使用这三种符号，尤其是在使用链式法则或进行隐函数求导时。
5. Power Rule and Basic Rules | 幂法则与基本法则
This is the power rule. It works for any real n, including negative and fractional powers
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