📚 Introduction to Differential Calculus | 微分学导论
Differential calculus is a fundamental branch of mathematics concerned with the study of rates of change. It gives us powerful tools to analyse how functions behave at any instant, enabling us to find slopes of curves, speeds of moving objects, and optimisation of real-world systems. The core idea is the derivative, which arises naturally from the geometric problem of drawing a tangent to a curve.
微分学是数学中一个基础的领域，专注于变化率的研究。它为我们提供了强大的工具，用于分析函数在任何瞬时的行为，使我们能够求出曲线的斜率、运动物体的速度以及现实世界中系统的最优化。其核心概念是导数，它自然地源于绘制曲线切线的几何问题。
1. What is Differential Calculus? | 什么是微分学？
Differential calculus, together with integral calculus, forms the discipline known as calculus. While integral calculus deals with accumulation of quantities and areas under curves, differential calculus focuses on instantaneous rates of change. The derivative of a function at a point tells us how sensitive the function’s output is to small changes in its input.
微分学与积分学共同构成了微积分这门学科。积分学研究量的累积和曲线下的面积，而微分学则关注瞬时变化率。函数在某点的导数告诉我们函数的输出对输入微小变化的敏感程度。
Imagine driving a car: the speedometer shows your instantaneous speed — the rate at which your position is changing with respect to time. This is exactly a derivative: the rate of change of distance with respect to time. In economics, derivatives describe marginal cost and marginal revenue; in biology, they model population growth rates.
想想开车的情景：速度表显示的是你的瞬时速度——即位置相对于时间的变化率。这正是导数：距离相对于时间的变化率。在经济学中，导数描述边际成本和边际收益；在生物学中，它们模拟种群增长速率。
2. Limits and Continuity | 极限与连续性
The derivative is built upon the concept of a limit. Informally, we say the limit of f(x) as x approaches a is L if the values of f(x) can be made arbitrarily close to L by taking x sufficiently close to a. We write this as limx→a f(x) = L. Limits allow us to handle behaviour very close to a point without necessarily evaluating the function exactly at that point.
导数建立在极限概念之上。非正式地，当 x 趋近于 a 时 f(x) 的极限是 L，意味着通过让 x 充分接近 a，可以使 f(x) 的值任意接近 L。我们将其记作 limx→a f(x) = L。极限使我们能够处理接近某点时的行为，而不必在该点处精确求值。
A function is continuous at a point a if the limit as x approaches a exists, the function value at a is defined, and they are equal. In calculus, we require functions to be continuous at points where we differentiate them, because abrupt jumps would prevent a meaningful rate of change from being defined.
如果当 x 趋近于 a 时极限存在、函数在 a 有定义且两者相等，则函数在 a 点连续。在微积分中，我们要求函数在我们要微分的点处是连续的，因为跳跃间断会阻止定义一个有意义的变化率。
3. The Derivative as a Limit | 导数作为极限
The derivative of a function f at a point x is defined as the limit of the average rate of change. Geometrically, we consider the slope of a secant line passing through (x, f(x)) and a nearby point (x+h, f(x+h)). As h approaches 0, the secant line approaches the tangent line, and its slope approaches the derivative.
函数 f 在 x 点的导数被定义为平均变化率的极限。从几何角度看，我们考虑通过 (x, f(x)) 和邻近点 (x+h, f(x+h)) 的割线的斜率。当 h 趋近于 0 时，割线趋近于切线，其斜率也就趋近于导数。
f'(x) = limh→0 [f(x+h) − f(x)] / h
This definition is called the first-principles definition of the derivative. It captures the instantaneous rate of change of f with respect to x. If this limit exists, we say f is differentiable at x.
这个定义称为导数的第一原理定义。它刻画了 f 关于 x 的瞬时变化率。如果该极限存在，我们就说 f 在 x 处可导。
4. Notation for Derivatives | 导数的记法
There are several common notations for derivatives, each useful in different contexts. The prime notation f'(x) was introduced by Lagrange and is read as “f prime of x”. Leibniz notation dy/dx reminds us that the derivative is a ratio of infinitesimal changes. In operator form, we write d/dx (f(x)). For specific values, we write f'(a) or dy/dx|x=a.
导数有几种常用的记法，每种在不同的语境下用起来很方便。撇号记法 f'(x) 由拉格朗日引入，读作 “f prime of x”。莱布尼茨记法 dy/dx 提醒我们导数是一个无穷小变化的比值。在算子形式中，我们写 d/dx (f(x))。对于特定值，我们写 f'(a) 或 dy/dx|x=a。
Second derivatives are denoted f”(x), d²y/dx², or d²/dx² (f(x)). Higher-order derivatives follow the same pattern. The Leibniz notation is particularly useful when applying the chain rule, as it clearly shows which variable is being differentiated with respect to.
二阶导数记作 f”(x)、d²y/dx² 或 d²/dx² (f(x))。更高阶的导数沿用同样的模式。莱布尼茨记法在应用链式法则时尤其有用，因为它清楚地表明了是对哪一个变量求导。
5. Basic Differentiation Rules | 基本求导法则
Before tackling complicated functions, it is essential to learn the elementary rules that allow us to calculate derivatives efficiently. These rules are derived from the limit definition and enable us to combine simpler derivatives.
在对付复杂的函数之前，有必要学习那些能让我们高效计算导数的基本法则。这些法则由极限定义推导而来，允许我们组合较简单的导数。
Constant rule: if f(x) = c, where c is a constant, then f'(x) = 0. Sum rule: the derivative of a sum is the sum of the derivatives, (f+g)’ = f’ + g’. Constant multiple rule: the derivative of a constant times a function is the constant times the derivative, (c f)’ = c f’.
常数法则：如果 f(x) = c，其中 c 为常数，则 f'(x) = 0。和差法则：和的导数等于导数的和，(f+g)’ = f’ + g’。常数倍法则：常数乘函数的导数等于常数乘函数的导数，(c f)’ = c f’。
6. The Power Rule | 幂法则
The power rule is one of the most frequently used rules in differential calculus. It states that for any real number n, the derivative of xⁿ is n xⁿ⁻¹. This rule can be proved for positive integer exponents using the binomial theorem and the limit definition.
幂法则是微分学中最常用的法则之一。它指出，对于任意实数 n，xⁿ 的导数为 n xⁿ⁻¹。对于正整数指数，可以用二项式定理和极限定义来证明该法则。
Examples: d/dx (x²) = 2x, d/dx (x³) = 3x², d/dx (x⁻¹) = −x⁻² = −1/x², and d/dx (√x) = d/dx (x½) = (½) x⁻½ = 1/(2√x). The power rule works for negative and fractional exponents as well, greatly simplifying differentiation of roots and reciprocals.
示例：d/dx (x²) = 2x，d/dx (x³) = 3x²，d/dx (x⁻¹) = −x⁻² = −1/x²，以及 d/dx (√x) = d/dx (x½) = (½) x⁻½ = 1/(2√x)。幂法则同样适用于负指数和分数指数，大大简化了方根和倒数的求导。
7. Derivatives of Polynomials | 多项式的导数
A polynomial is a sum of constant multiples of power functions. Using the sum rule, constant multiple rule, and the power rule, we can differentiate any polynomial term by term. For instance, if f(x) = 4x⁵ − 3x² + 2x − 7, then f'(x) = 20x⁴ − 6x + 2.
多项式是一些常数倍的幂函数之和。使用和差法则、常数倍法则以及幂法则，我们可以逐项地对任何多项式求导。比如，如果 f(x) = 4x⁵ − 3x² + 2x − 7，那么 f'(x) = 20x⁴ − 6x + 2。
This term-by-term differentiation is straightforward because the derivative operator is linear: it distributes over addition and respects scalar multiplication. The process reduces the degree of each non-constant term by one, which is why differentiating a cubic yields a quadratic, and so on.
这种逐项求导之所以直接，是因为导数算子是线性的：它对加法满足分配律，并与数乘相容。该过程将每个非常数项的次数减一，这就是为什么对三次多项式求导会得到一个二次多项式，如此等等。
8. Derivatives of Trigonometric Functions | 三角函数的导数
The derivatives of the sine and cosine functions are intimately linked. Using the limit definition and the well-known limit limh→0 (sin h)/h = 1, one can show that d/dx (sin x) = cos x and d/dx (cos x) = −sin x. From these, the derivative of tan x follows by the quotient rule: d/dx (tan x) = sec² x.
正弦和余弦函数的导数密切相关。利用极限定义和熟知的极限 limh→0 (sin h)/h = 1，可以证明 d/dx (sin x) = cos x 以及 d/dx (cos x) = −sin x。由此，通过商法则可得到 tan x 的导数：d/dx (tan x) = sec² x。
These results reveal the periodic nature of trigonometric derivatives: differentiating sine gives cosine, differentiating cosine gives negative sine, and differentiating negative sine gives negative cosine, returning to sine after four derivatives. This cyclic behaviour is useful when dealing with higher-order derivatives.
这些结果揭示了三角函数导数的周期性：正弦求导得余弦，余弦求导得负正弦，负正弦求导得负余弦，经过四次求导后又回到正弦。这种循环特性在处理高阶导数时十分有用。
9. The Product and Quotient Rules | 乘积法则与商法则
When differentiating products or quotients of two functions, we cannot simply differentiate each part separately. The product rule states: if y = u v, where u and v are functions of x, then dy/dx = u’ v + u v’. In prime notation, (u v)’ = u’ v + u v’.
当对两个函数的乘积或商求导时，我们不能简单地对各部分分别求导。乘积法则指出：如果 y = u v，其中 u 和 v 是 x 的函数，那么 dy/dx = u’ v + u v’。用撇号记法就是 (u v)’ = u’ v + u v’。
The quotient rule deals with ratios. For y = u/v, the derivative is dy/dx = (u’ v − u v’) / v². A common memory aid is “low d-high minus high d-low, over low squared”. It is important to keep the subtraction order, because the derivative is not symmetric.
商法则处理比值的情况。对于 y = u/v，导数为 dy/dx = (u’ v − u v’) / v²。一个常用的记忆口诀是 “分母乘分子的导数减去分子乘分母的导数，除以分母的平方”。注意保持减法的顺序，因为导数并不对称。
Example: if y = x² sin x, then by the product rule, y’ = 2x sin x + x² cos x. For y = (x³ + 1)/(x − 2), the quotient rule gives y’ = [(3x²)(x−2) − (x³+1)(1)] / (x−2)².
示例：如果 y = x² sin x，那么根据乘积法则 y’ = 2x sin x + x² cos x。对于 y = (x³ + 1)/(x − 2)，应用商法则得到 y’ = [(3x²)(x−2) − (x³+1)(1)] / (x−2)²。
10. The Chain Rule | 链式法则
The chain rule is used to differentiate composite functions — functions of the form y = f(g(x)). It expresses the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function. In Leibniz notation: dy/dx = dy/du · du/dx, where u = g(x).
链式法则用于对复合函数求导，即形如 y = f(g(x)) 的函数。它表示外层函数对里层函数的导数乘上里层函数的导数。用莱布尼茨记法：dy/dx = dy/du · du/dx，其中 u = g(x)。
For example, if y = (3x + 2)⁴, let u = 3x + 2, then y = u⁴. dy/du = 4u³, du/dx = 3, so dy/dx = 4u³ · 3 = 12(3x+2)³. As you become proficient, you can apply the chain rule mentally by differentiating the outer power, keeping the inside unchanged, and multiplying by the derivative of the inside.
例如，若 y = (3x + 2)⁴，令 u = 3x + 2，则 y = u⁴。dy/du = 4u³，du/dx = 3，因此 dy/dx = 4u³ · 3 = 12(3x+2)³。熟练后，你可以心算应用链式法则：对外层幂函数求导，保持内部不变，再乘上内部的导数。
The chain rule also applies to trigonometric compositions, such as y = sin(5x) gives y’ = 5 cos(5x), and y = cos(x²) gives y’ = −2x sin(x²). It is arguably the most important differentiation rule, required whenever functions are nested.
链式法则也适用于三角函数的复合，例如 y = sin(5x) 的导数为 y’ = 5 cos(5x)，y = cos(x²) 的导数为 y’ = −2x sin(x²)。它可以说是最重要的求导法则，只要遇上有函数嵌套的情况就离不开它。
11. Higher-Order Derivatives | 高阶导数
After finding the first derivative, we can differentiate again to obtain the second derivative f”(x) or d²y/dx². The second derivative measures the rate of change of the rate of change. In kinematics, if position is s(t), then velocity is s'(t) and acceleration is s”(t).
求得一阶导数后，我们可以再次求导以得到二阶导数 f”(x) 或 d²y/dx²。二阶导数衡量的是变化率的变化率。在运动学中，如果位移是 s(t)，那么速度就是 s'(t)，加速度就是 s”(t)。
Higher-order derivatives are defined recursively: the third derivative is the derivative of the second derivative, and so on. For a polynomial like f(x) = x⁴, we have f'(x) = 4x³, f”(x) = 12x², f”'(x) = 24x, f””(x) = 24, and the fifth derivative is 0. The second derivative also tells us about the concavity of a graph.
高阶导数通过递推定义：三阶导数是二阶导数的导数，依此类推。对于多项式 f(x) = x⁴，我们有 f'(x) = 4x³，f”(x) = 12x²，f”'(x) = 24x，f””(x) = 24，而五阶导数为 0。二阶导数还能告诉我们函数图像的凹凸性。
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