📚 Finding the Derivative | 求导数
the curve at any point, that is, the instantaneous rate at which y changes with respect to x. The formal definition from first principles, f′(x) = lim (h → 0) [ f(x + h) − f(x) ] / h, is the foundation on which all differentiation rules are built, and it can appear in Edexcel examination questions.
求导是微分学的核心运算。如果 y = f(x)，那么导数 dy/dx 或 f′(x) 表示曲线在任意点处的斜率，也就是 y 相对于 x 的瞬时变化率。基于第一原理（定义）的正式定义为 f′(x) = lim (h → 0) [ f(x + h) − f(x) ] / h，它是所有求导法则的基础，也可能出现在 Edexcel 考试题中。
1. What is the Derivative? | 什么是导数？
ells you how sensitive the output is to a small change in the input. Geometrically, as the two points used to calculate a chord get closer together, the gradient of the chord approaches the gradient of the tangent.
函数 f 在点 x 处的导数是曲线 y = f(x) 在该点处切线的斜率。它表示当输入发生微小变化时，输出变化的敏感程度。从几何上看，当用来计算弦的两个点越来越接近时，弦的斜率就趋近于切线的斜率。
f′(x) = lim (h → 0) [ f(x + h) − f(x) ] / h
Δy / Δx is the average rate of change over an interval, while dy/dx is the instantaneous rate of change at a single point. A derivative can be thought of as a function in its own right: for each x where the limit exists, it returns the gradient at that x.
使用莱布尼茨记号，我们写作 dy/dx = lim (Δx → 0) Δy / Δx。其中 Δy / Δx 是区间上的平均变化率，而 dy/dx 是某一点处的瞬时变化率。导数本身也可以看作一个函数：在极限存在的每一个 x 处，它给出该点的斜率。
2. The Formal Definition from First Principles | 基于第一原理的正式定义
The definition from first principles is a limit. To differentiate f(x) from first principles, we form the difference quotient and then let h tend to zero. The method works for any differentiable function, although the algebra can be lengthy.
基于第一原理的定义是一个极限。要对 f(x) 用第一原理求导，我们先构造差商，然后让 h 趋于零。这个方法适用于任何可导函数，尽管代数运算有时会比较长。
- Find f(x + h) and subtract f(x). | 求出 f(x + h)，再减去 f(x)。
- Divide the result by h to form [ f(x + h) − f(x) ] / h. | 将结果除以 h，得到 [ f(x + h) − f(x) ] / h。
- Simplify the expression, cancelling h where possible. | 化简表达式，尽可能约去 h。
- Take the limit as h → 0. | 令 h → 0，取极限。
The derivative exists only if this limit exists. If the left-hand limit and the right-hand limit are different, or if the limit is infinite, then the function is not differentiable at that point. In Edexcel papers, a first-principles question usually asks you to prove a given result or to differentiate a simple polynomial or reciprocal function.
只有当这个极限存在时，导数才存在。如果左极限与右极限不同，或者极限为无穷大，那么函数在该点不可导。在 Edexcel 试卷中，第一原理题目通常要求你证明某个已知结果，或对简单的多项式、倒数函数求导。
3. Notation: Leibniz, Lagrange and d/dx | 导数记号：莱布尼茨、拉格朗日与 d/dx
Several notations are used for the same idea, and exam questions may switch between them. You must be able to read and write all of the common forms fluently.
同一个概念有多种记号，考试题目可能会在不同记号之间切换。你必须能够熟练地读写常见的几种形式。
| Notation | Meaning | 中文说明 |
| f′(x) | Lagrange notation; derivative of f with respect to x | 拉格朗日记号；f 对 x 的导数 |
| dy/dx | Leibniz notation; derivative of y with respect to x | 莱布尼茨记号；y 对 x 的导数 |
| d/dx [ f(x) ] | Operator notation; differentiate the expression with respect to x | 算子记号；对括号内表达式关于 x 求导 |
| f′(a) | The value of the derivative at x = a | 导数在 x = a 处的值 |
The symbol d/dx is an operator: it acts on whatever follows it. For example, d/dx (x² + 3x) means differentiate x² + 3x with respect to x. In mechanics, a dot notation such as ẋ is sometimes used for a derivative with respect to time, but this is less common in pure mathematics papers.
符号 d/dx 是一个算子：它对后面跟的表达式起作用。例如，d/dx (x² + 3x) 表示对 x² + 3x 关于 x 求导。在力学中，有时会用点记号如 ẋ 表示对时间的导数，但在纯数学试卷中不太常见。
4. Derivative as Gradient and Rate of Change | 导数作为斜率与变化率
s the instantaneous rate of change of y with respect to x. For example, if s(t) is displacement, then ds/dt is velocity, and the second derivative d²s/dt² is acceleration.
导数有两个主要解释。几何上，f′(a) 是曲线 y = f(x) 在 x = a 处切线的斜率。在实际意义上，它是 y 相对于 x 的瞬时变化率。例如，如果 s(t) 表示位移，那么 ds/dt 是速度，二阶导数 d²s/dt² 是加速度。
Average rate of change = Δy / Δx = [ f(x + Δx) − f(x) ] / Δx
e instantaneous rate of change, which is the derivative. This distinction between average and instantaneous is a common source of explanation marks in structured questions.
区间上的平均变化率就是弦的斜率。令 Δx → 0 取极限，就得到瞬时变化率，即导数。平均变化率与瞬时变化率的区别，是结构化题目中常见的得分点。
5. Differentiability, Continuity and Non-differentiable Points | 可导性、连续性与不可导点
If a function is differentiable at a point, then it must be continuous at that point. However, continuity does not guarantee differentiability. A function can be continuous but have a corner, a cusp, or a vertical tangent, in which case the derivative may fail to exist.
如果函数在某点可导，那么它在该点必定连续。但连续并不能保证可导。函数可以连续，却存在尖角、尖点或竖直切线，此时导数可能不存在。
- f(x) = |x| t the left-hand gradient is −1 and the right-hand gradient is +1, so f′(0) does not exist. | f(x) = |x| 在 x = 0 处连续，但左导数为 −1，右导数为 +1，因此 f′(0) 不存在。
- f(x) = x1/3 has a vertical tangent at x = 0, so the derivative tends to infinity. | f(x) = x1/3 在 x = 0 处有竖直切线，导数趋于无穷大。
- A step function is not continuous, so it cannot be differentiable at the jump. | 阶梯函数不连续，因此在跳跃点处不可导。
For piecewise functions, you should check the left-hand and right-hand limits of the difference quotient separately. Only when both limits exist and are equal is the function differentiable at that point.
对于分段函数，应分别检查差商的左极限与右极限。只有当两个极限都存在且相等时，函数在该点才可导。
6. Differentiation from First Principles: Worked Examples | 用第一原理求导：例题
ting f(x) = x² gives 2xh + h². Dividing by h gives 2x + h. Letting h → 0 gives dy/dx = 2x.
例 1：用第一原理求 y = x² 的导数。先写出 f(x + h) = (x + h)² = x² + 2xh + h²。减去 f(x) = x²，得到 2xh + h²。除以 h，得到 2x + h。令 h → 0，得到 dy/dx = 2x。
y = x² ⇒ dy/dx = 2x
h / [ x(x + h) ]. Dividing by h gives −1 / [ x(x + h) ]. Letting h → 0 gives dy/dx = −1/x².
例 2：用第一原理求 y = 1/x 的导数，其中 x ≠ 0。有 f(x + h) = 1/(x + h)。差为 1/(x + h) − 1/x = [ x − (x + h) ] / [ x(x + h) ] = −h / [ x(x + h) ]。除以 h 得 −1 / [ x(x + h) ]。令 h → 0，得到 dy/dx = −1/x²。
y = 1/x ⇒ dy/dx = −1/x²
Example 3: For y = √x, multiply numerator and denominator by the conjugate √x + h + √x. The difference quotient becomes 1 / [ √x + h + √x ], and as h → 0 this tends to 1 / (2√x). These three results are worth remembering because they illustrate the standard algebraic techniques used in first-principles questions.
例 3：对于 y = √x，将分子分母同乘共轭式 √x + h + √x。差商变为 1 / [ √x + h + √x ]，当 h → 0 时趋于 1 / (2√x)。这三个结果值得记住，因为它们展示了第一原理题目中常用的代数技巧。
7. Power Rule and Polynomials | 幂法则与多项式
The power rule is the most frequently used differentiation rule. For any real number n, the derivative of xⁿ is n xⁿ⁻¹. This rule also works for negative and fractional indices, provided the function is defined and differentiable at the point considered.
幂法则是使用最频繁的求导法则。对任意实数 n，xⁿ 的导数是 n xⁿ⁻¹。该法则也适用于负指数和分数指数，只要函数在所考虑的点有定义且可导。
d/dx (xⁿ) = n xⁿ⁻¹
- d/dx (x⁵) = 5x⁴ | d/dx (x⁵) = 5x⁴
- d/dx (x⁻²) = −2x⁻³ = −2/x³ | d/dx (x⁻²) = −2x⁻³ = −2/x³
- d/dx (√x) = d/dx (x^½) = ½ x−½ = 1/(2√x) | d/dx (√x) = d/dx (x^½) = ½ x−½ = 1/(2√x)
ero, and the derivative of a sum is the sum of the derivatives. For example, d/dx (3x⁴ − 5x² + 7x − 2) = 12x³ − 10x + 7.
结合常数法则与和法则，幂法则可以让你逐项求导任何多项式。常数的导数为零，和的导数等于导数的和。例如，d/dx (3x⁴ − 5x² + 7x − 2) = 12x³ − 10x + 7。
8. Trigonometric, Exponential and Logarithmic Derivatives | 三角、指数与对数函数的导数
Standard derivatives of trigonometric, exponential and logarithmic functions must be memorised. In particular, the trigonometric derivatives are valid only when angles are measured in radians, which is the default in calculus.
三角、指数与对数函数的标准导数必须记住。特别要注意，三角函数的导数公式只在角度用弧度制时成立，而弧度制是微积分中的默认单位。
| Function | Derivative | 中文说明 |
| sin x | cos x | 正弦的导数为余弦 |
| cos x | −sin x | 余弦的导数为负正弦 |
| tan x | sec² x | 正切的导数为正割平方 |
| eˣ | eˣ | 指数函数 eˣ 的导数是自身 |
| ln x | 1/x | 自然对数的导数为 1/x |
| aˣ | aˣ ln a | 一般指数函数的导数 |
s combine with the sum, product, quotient and chain rules. For example, d/dx (3 sin x + 2eˣ − ln x) = 3 cos x + 2eˣ − 1/x. It is also useful to remember that
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