📚 Differentiation: Definitions and Rules | 微分：定义与法则
Differentiation lies at the heart of calculus. It allows us to describe how a quantity changes instantaneously – the slope of a curve, the velocity of a moving object, or the marginal cost in economics. This article covers the fundamental definition of the derivative as a limit and the essential rules that make differentiation efficient. Mastering these definitions and rules is the first step towards solving real-world problems involving rates of change.
微分是微积分的核心。它让我们能够描述一个量如何瞬间变化——曲线的斜率、运动物体的速度，或者经济学中的边际成本。本文涵盖了导数作为极限的基本定义，以及使微分运算高效的关键法则。掌握这些定义和法则是解决涉及变化率的实际问题的重要第一步。
1. The Derivative as a Limit | 作为极限的导数
The derivative of a function f at a point x is defined formally as:
函数 f 在点 x 处的导数被正式定义为：
f '(x) = limh→0 (f(x + h) – f(x)) / h
the graph of y = f(x) at that point. If the limit exists, we say f is differentiable at x.
只要该极限存在。从几何上看，这个极限给出了曲线 y = f(x) 在该点处切线的斜率。如果极限存在，我们就说 f 在 x 点可导。
The expression (f(x+h) – f(x)) / h is called the difference quotient. As h approaches zero, the secant line through two nearby points becomes the tangent line. This definition links algebra and geometry, forming the foundation for all differentiation techniques.
表达式 (f(x+h) – f(x)) / h 被称为差商。当 h 趋近于零时，穿过两个相邻点的割线就变成了切线。这一定义将代数与几何联系起来，为所有的微分技巧奠定了基础。
2. Notation | 记号
Several notations are used for the derivative of y = f(x). The most common are Lagrange's notation f '(x) and Leibniz's notation dy/dx. Another popular form is the operator notation d/dx f(x).
对 y = f(x) 的导数有几种不同的记号。最常用的是拉格朗日记号 f '(x) 和莱布尼茨记号 dy/dx。另一种常见形式是算符记号 d/dx f(x)。
When evaluating the derivative at a specific point x = a, we write f '(a) or dy/dx|x=a. Leibniz's notation is particularly useful because it explicitly shows the variable with respect to which we are differentiating, and it simplifies later work with the chain rule.
当在特定点 x = a 处计算导数时，我们写成 f '(a) 或 dy/dx|x=a。莱布尼茨记号特别有用，因为它明确显示了我们是对哪个变量求导，并且在以后使用链式法则时更为简便。
3. The Power Rule | 幂法则
For any real constant n, the derivative of xn is given by:
对于任意实数常量 n，xn 的导数由下式给出：
d/dx (xn) = n xn – 1
For example, d/dx (x3) = 3x2 and d/dx (x½) = ½ x–½. The power rule works for positive integers, negative integers, fractions, and even irrational exponents, as long as the function is differentiable on its domain.
例如，d/dx (x3) = 3x2；以及 d/dx (x½) = ½ x–½。幂法则对正整数、负整数、分数甚至无理指数都适用，只要函数在其定义域上可导。
Remember that before applying the power rule, expressions should be written as x to a power. For instance, 1/x2 becomes x–2, and √x becomes x½. The rule is one of the most frequently used shortcuts in differentiation.
记住，在应用幂法则之前，应先将表达式写成 x 的幂次形式。例如，1/x2 要写成 x–2，而 √x 要写成 x½。该法则是最常用的微分快捷方式之一。
4. Constant Rule and Constant Multiple Rule | 常数法则与常数倍法则
zero: d/dx (c) = 0, because the graph of a constant is a horizontal line with zero slope.
常数函数 c 的导数为零：d/dx (c) = 0，因为常数的图形是一条斜率为零的水平线。
If a function is multiplied by a constant, the derivative is multiplied by that same constant:
如果一个函数乘以一个常数，其导数也乘以同一个常数：
d/dx [c · f(x)] = c · f '(x)
For example, d/dx (5x4) = 5 · 4x3 = 20x3. These two simple rules allow us to handle any linear combination of terms straightforwardly.
例如，d/dx (5x4) = 5 · 4x3 = 20x3。这两条简单规则使我们能够直接处理任何项的线性组合。
5. Sum and Difference Rules | 和差法则
The derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives:
和的导数是导数的和，差的导数是导数的差：
d/dx [f(x) ± g(x)] = f '(x) ± g '(x)
Combined with the constant multiple rule, this allows us to differentiate any polynomial term by term. For example, d/dx (3x5 – 2x3 + 7x – 4) = 15x4 – 6x2 + 7.
与常数倍法则结合，这使得我们可以对任何多项式逐项求导。例如，d/dx (3x5 – 2x3 + 7x – 4) = 15x4 – 6x2 + 7。
These linearity properties of the derivative are fundamental and can be proven directly from the limit definition. They make differentiation a very structured and predictable operation.
导数的这些线性性质是基本的，可以直接从极限定义证明。这使得微分成为一种非常有条理且可预测的运算。
6. The Product Rule | 积法则
When differentiating a product of two functions, we cannot simply multiply the derivatives. Instead, we use:
对两个函数的乘积求导时，不能简单地将导数相乘。相反，我们使用：
d/dx [u(x) v(x)] = u '(x) v(x) + u(x) v '(x)
A classic example: d/dx (x2 sin x) = 2x sin x + x2 cos x. The product rule is essential whenever the function is expressed as two distinct factors whose derivatives are easier to compute separately.
一个经典例子：d/dx (x2 sin x) = 2x sin x + x2 cos x。当一个函数可以表示为两个独立因式的乘积，且它们各自的导数更容易计算时，积法则就变得至关重要。
It is good practice to factor the result after applying the product rule, as this often reveals critical points or simplifies further work. Remember that the order does not matter: u'v + uv' is symmetric.
最好在应用积法则后将结果因式分解，因为这常常能揭示关键点或简化后续运算。记住顺序并不重要：u'v + uv' 是对称的。
7. The Quotient Rule | 商法则
For a quotient of two functions y = u(x) / v(x), where v(x) ≠ 0, the derivative is:
对于两个函数的商 y = u(x) / v(x)，其中 v(x) ≠ 0，导数为：
d/dx (u/v) = (u ' v – u v ') / v2
For instance, d/dx (tan x) = (sec2 x · cos x – sin x · (–sin x)) / cos2 x, which simplifies to sec2 x. The quotient rule is useful, but sometimes it is easier to rewrite the function as a product with a negative exponent and then use the product rule or chain rule.
例如，d/dx (tan x) = (sec2 x · cos x – sin x · (–sin x)) / cos2 x，可化简为 sec2 x。商法则很有用，但有时将函数重写为带有负指数的乘积形式，然后使用积法则或链式法则会更容易。
A common memory aid is "low d high minus high d low, over low squared," where "low" is the denominator v and "high" is the numerator u. Be careful with the subtraction order: it is u'v – uv', not the other way around.
一个常见的记忆口诀是“下面乘上面导减上面乘下面导，除以下面平方”，其中“下面”指分母 v，“上面”指分子 u。注意减法的顺序：是 u'v – uv'，而不是反过来。
8. The Chain Rule | 链式法则
i.e., functions of the form f(g(x)). In Leibniz notation, if y = f(u) and u = g(x), then:
链式法则用于求复合函数的导数，即形如 f(g(x)) 的函数。用莱布尼茨记号表示，如果 y = f(u) 且 u = g(x)，那么：
dy/dx = (dy/du) × (du/dx)
In prime notation: [f(g(x))] ' = f '(g(x)) · g '(x). For example, d/dx (sin(x3)) = cos(x3) · 3x2. The chain rule extends to multiple layers of composition, working from the outside in.
用“prime”记号表示为：[f(g(x))] ' = f '(g(x)) · g '(x)。例如，d/dx (sin(x3)) = cos(x3) · 3x2。链式法则可以延伸到多层复合函数，从外向内逐层求导。
Mastering the chain rule is vital because it combines with all other differentiation rules. Many student errors come from forgetting to multiply by the derivative of the inner function. Always check for an "inner" function before applying the basic rules.
掌握链式法则至关重要，因为它与所有其他微分法则结合在一起。许多学生的错误源自忘记乘以内层函数的导数。在应用基本法则之前，务必检查是否存在“内层”函数。
9. Derivatives of Trigonometric Functions | 三角函数的导数
When x is measured in radians, the derivatives of the basic trigonometric functions are:
- d/dx (sin x) = cos x
- d/dx (cos x) = –sin x
- d/dx (tan x) = sec2 x
These can be proven using the limit definition and the angle addition formulas. The reciprocal functions are also common:
当 x 以弧度为单位时，基本三角函数的导数为：
- d/dx (sin x) = cos x
- d/dx (cos x) = –sin x
- d/dx (tan x) = sec2 x
这些都可以用极限定义与角的和差公式证明。它们的倒数函数也很常见：
- d/dx (sec x) = sec x tan x
- d/dx (csc x) = –csc x cot x
- d/dx (cot x) = –csc2 x
IB courses require these derivatives in radian mode; working in degrees introduces awkward constants. Combined with the chain rule, you can differentiate functions like sin(2x) easily: derivative is 2 cos(2x).
IB 课程要求这些导数在弧度制下使用；用角度制会引入繁琐的常数。结合链式法则，你可以轻松地对诸如 sin(2x) 这样的函数求导：导数为 2 cos(2x)。
10. Derivatives of Exponential and Logarithmic Functions | 指数函数与对数函数的导数
The natural exponential function ex is its own derivative:
自然指数函数 ex 的导数就是它本身：
d/dx (ex) = ex
For an exponential with base a > 0, d/dx (ax) = ax ln a. For the natural logarithm:
对于底数 a > 0 的指数函数，d/dx (ax) = ax ln a。对于自然对数：
d/dx (ln x) = 1/x, for x > 0
For a general logarithm, d/dx (loga x) = 1 / (x ln a). These formulas, especially for ex and ln x, appear constantly in modeling and solving differential equations.
对于一般对数，d/dx (loga x) = 1 / (x ln a)。这些公式，特别是 ex 和 ln x 的导数，在建模和求解微分方程中频繁出现。
When differentiating expressions like e3x or ln(5x), the chain rule is required: d/dx (e3x) = 3e3x, and d/dx (ln(5x)) = 1/x (since ln(5x) = ln5 + lnx).
当对 e3x 或 ln(5x) 这样的表达式求导时，需要使用链式法则：d/dx (e3x) = 3e3x；d/dx (ln(5x)) = 1/x（因为 ln(5x) = ln5 + lnx）。
11. Higher-Order Derivatives | 高阶导数
Taking the derivative repeatedly gives higher-order derivatives. The second derivative, denoted f ''(x) or d2y/dx2, measures the rate of change of the first derivative; it describes concavity and acceleration.
反复求导会得到高阶导数。二阶导数记作 f ''(x) 或 d2y/dx2，它衡量一阶导数的变化率；描述了曲线的凹性和加速度。
If s(t) is a position function, then s '(t) is velocity v(t), and s ''(t) is acceleration a(t). In Leibniz notation, the second derivative is written as d2y/dx2, not d2y/d2x, which is a common mistake.
如果 s(t) 是位置函数，那么 s '(t) 是速度 v(t)，而 s ''(t) 是加速度 a(t)。用莱布尼茨记号写二阶导数时，应写作 d2y/dx2，而不是 d2y/d2x，这是一个常见错误。
Higher-order derivatives are also used in Taylor polynomials and in understanding the behavior of functions. The third derivative is sometimes called "jerk." The notation f (n)(x) is used for the n-th derivative when n ≥ 3.
高阶导数也用于泰勒多项式以及理解函数的形态。三阶导数有时被称为“加加速度”。当 n ≥ 3 时，常用 f (n)(x) 表示 n 阶导数。
12. Implicit Differentiation | 隐函数微分
on is not given explicitly as y = f(x) but rather defined implicitly by an equation like x2 + y2 n. We differentiate both sides with respect to x, treating y as a function of x and applying the chain rule.
当一个函数并非以显式 y = f(x) 给出，而是通过诸如 x2 + y2 = 25 的方程隐式定义时，我们使用隐函数微分。我们对等式两边关于 x 求导，将 y 视为 x 的函数并应用链式法则。
Differentiating x2 + y2 = 25 gives 2x + 2y (dy/dx) = 0, so dy/dx = – n curves like circles and ellipses, where solving for y explicitly is messy or impossible.
对 x2 + y2 = 25 求导，得到 2x + 2y (dy/dx) = 0，所以 dy/dx = –x/y。这一技巧在求圆和椭圆等曲线的切线斜率时特别有效，因为在那类曲线上显式解出 y 往往非常繁琐或不可行。
Implicit differentiation also extends to related rates problems, where we differentiate an equation linking several variables with respect to time t. This approach is a direct application of the chain rule in a multivariable context, though presented as a single-variable technique in IB.
隐函数微分还延伸到相关变化率问题，在那类问题中，我们对一个联系多个变量的方程关于时间 t 求导。这种方法是在多变量情境下对链式法则的直接应用，不过在 IB 课程中是作为单变量技巧讲授的。
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