📚 IB Mathematics: Differentiation Formulas and Computation | IB数学：求导公式与计算
The derivative measures how a function changes as its input changes. In IB Mathematics, you are expected to differentiate a wide range of functions fluently, both by using standard formulas and by applying the correct rule in composite or combined functions. The derivative \( f'(x) \) is defined as the limit below, when this limit exists.
导数衡量函数随输入量变化的快慢。在IB数学中，学生需要熟练地对各种函数求导，既能使用标准公式，也要在复合函数和组合函数中选用正确法则。导数 \( f'(x) \) 定义为下面的极限（若极限存在）：
f'(x) = lim (h→0) [f(x + h) – f(x)] / h
This limit definition is rarely used for direct computation in the exam, but it explains where all the differentiation rules come from. Once you understand the notation, you can apply the algebraic rules with speed and confidence.
这一极限定义在考试中很少用于直接计算，但它解释了所有求导法则的来源。只要理解了导数符号的含义，你就能快速、自信地应用各种代数法则。
1. Basic Differentiation Rules | 1. 基本求导法则
Before handling complex functions, you must master the simplest rules. The constant rule says that the derivative of any constant is zero. The constant multiple rule allows you to pull a constant out in front. The sum and difference rules let you differentiate term by term.
在处理复杂函数之前，必须先掌握最简单的法则。常数法则指出任意常数的导数为零；常数倍法则允许将常数提到导数符号前面；和差法则允许逐项求导。
d/dx (c) = 0, d/dx [c × f(x)] = c × f'(x), d/dx [u ± v] = u’ ± v’
For example, if \( y = 3x^2 + 5x – 2 \), then differentiating term by term gives \( y’ = 6x + 5 \).
例如，若 \( y = 3x^2 + 5x – 2 \)，逐项求导可得 \( y’ = 6x + 5 \)。
2. The Power Rule | 2. 幂函数求导法则
The power rule is one of the most frequently used formulas in IB differentiation. It states that for any real number \( n \), the derivative of \( x^n \) is \( n x^{n-1} \).
幂函数求导法则是IB求导中最常用的公式之一。对于任意实数 \( n \)，\( x^n \) 的导数为 \( n x^{n-1} \)。
d/dx (xⁿ) = n xⁿ⁻¹
The rule works for negative and fractional exponents too. For example, \( f(x) = 1/x^2 = x^{-2} \) has derivative \( f'(x) = -2x^{-3} = -2/x^3 \).
该法则同样适用于负指数和分数指数。例如 \( f(x) = 1/x^2 = x^{-2} \) 的导数为 \( f'(x) = -2x^{-3} = -2/x^3 \)。
For square roots, rewrite \( \sqrt{x} \) as a power before differentiating: if \( y = \sqrt{x} \), then \( y’ = 1/(2\sqrt{x}) \). This approach is essential when differentiating any radical expression.
对于平方根，求导前先将 \( \sqrt{x} \) 写成幂的形式：若 \( y = \sqrt{x} \)，则 \( y’ = 1/(2\sqrt{x}) \)。这种方法对任何根式表达式都至关重要。
3. Exponential and Logarithmic Functions | 3. 指数函数与对数函数
Exponential and logarithmic functions appear throughout the IB syllabus. Their derivatives follow specific formulas that must be committed to memory.
指数函数和对数函数贯穿IB课程始终，其导数公式需要牢记。
- \( d/dx(e^x) = e^x \). The natural exponential function is its own derivative.
- \( d/dx(a^x) = a^x \ln a \), for \( a > 0, a \neq 1 \).
- \( d/dx(\ln x) = 1/x \), for \( x > 0 \).
- \( d/dx(\log_a x) = 1/(x \ln a) \), for \( a > 0, a \neq 1 \).
For example, if \( y = 3^x \), then \( y’ = 3^x \ln 3 \). If \( y = \log_{10} x \), then \( y’ = 1/(x \ln 10) \).
例如，若 \( y = 3^x \)，则 \( y’ = 3^x \ln 3 \)。若 \( y = \log_{10} x \)，则 \( y’ = 1/(x \ln 10) \)。
4. Trigonometric Functions | 4. 三角函数求导
All trigonometric derivative formulas in IB assume that angles are measured in radians. The derivatives of sine and cosine are the most essential, and the others can be derived from them using quotient and chain rules.
IB中所有三角函数导数公式均假设角以弧度为单位。正弦和余弦的导数最为基础，其他三角函数的导数可以通过商法则和链式法则推导。
| Function | Derivative |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| sec x | sec x tan x |
| cosec x (csc x) | -cosec x cot x |
| cot x | -cosec² x |
For example, \( y = \tan x \) at \( x = 0 \) has derivative \( y’ = \sec^2 0 = 1 \).
例如，\( y = \tan x \) 在 \( x = 0 \) 处的导数为 \( y’ = \sec^2 0 = 1 \)。
5. The Product Rule | 5. 乘积法则
When two functions are multiplied together, you cannot differentiate each factor separately and multiply the derivatives. The product rule provides the correct method.
当两个函数相乘时，不能分别对每个因子求导后再相乘。乘积法则给出了正确的方法。
d/dx (u × v) = u’ × v + u × v’
Example: differentiate \( y = x^2 \sin x \).
示例：求 \( y = x^2 \sin x \) 的导数。
Let \( u = x^2 \) and \( v = \sin x \). Then \( u’ = 2x \) and \( v’ = \cos x \). Therefore \( y’ = 2x \sin x + x^2 \cos x \).
令 \( u = x^2 \)，\( v = \sin x \)，则 \( u’ = 2x \)，\( v’ = \cos x \)。因此 \( y’ = 2x \sin x + x^2 \cos x \)。
6. The Quotient Rule | 6. 商法则
The quotient rule is used when one function is divided by another. It is similar to the product rule, but the order of terms matters because of the minus sign.
当一个函数除以另一个函数时使用商法则。它与乘积法则类似，但由于负号的出现，项的先后顺序非常重要。
d/dx (u / v) = (u’ × v – u × v’) / v²
Example: differentiate \( y = x / (x^2 + 1) \).
示例：求 \( y = x / (x^2 + 1) \) 的导数。
- Let \( u = x \), so \( u’ = 1 \).
- Let \( v = x^2 + 1 \), so \( v’ = 2x \).
- Then \( y’ = [1(x^2 + 1) – x(2x)] / (x^2 + 1)^2 = (1 – x^2)/(x^2 + 1)^2 \).
注意：当分母为常数时，无需使用商法则，直接改写为常数倍后求导更简便。
Note: if the denominator is a constant, you do not need the quotient rule; rewrite as a constant multiple and use simpler rules.
7. The Chain Rule | 7. 链式法则
Composite functions, or “functions of functions”, require the chain rule. This is one of the most important skills in IB differentiation because it appears in nearly every differentiation question.
复合函数，即“函数的函数”，需要使用链式法则。这是IB求导中最关键的技能之一，几乎出现在每一道求导题中。
d/dx f(g(x)) = f'(g(x)) × g'(x)
In Leibniz notation, if \( y = f(u) \) and \( u = g(x) \), then \( dy/dx = dy/du × du/dx \).
用莱布尼茨记号表示：若 \( y = f(u) \)，且 \( u = g(x) \)，则 \( dy/dx = dy/du × du/dx \)。
Example: differentiate \( y = (2x + 1)^3 \). Let \( u = 2x + 1 \). Then \( dy/du = 3u^2 \) and \( du/dx = 2 \). So \( dy/dx = 3(2x + 1)^2 × 2 = 6(2x + 1)^2 \).
示例：求 \( y = (2x + 1)^3 \) 的导数。令 \( u = 2x + 1 \)，则 \( dy/du = 3u^2 \)，\( du/dx = 2 \)。所以 \( dy/dx = 3(2x + 1)^2 × 2 = 6(2x + 1)^2 \)。
A second example: \( y = \sin(x^2 + 1) \). The derivative is \( \cos(x^2 + 1) × 2x = 2x \cos(x^2 + 1) \).
另一个例子：\( y = \sin(x^2 + 1) \)，其导数为 \( \cos(x^2 + 1) × 2x = 2x \cos(x^2 + 1) \)。
8. Higher Derivatives | 8. 高阶导数
The second derivative, written \( f”(x) \) or \( d^2y/dx^2 \), is the derivative of the derivative. Higher derivatives are used in convexity analysis and kinematics.
二阶导数，记作 \( f”(x) \) 或 \( d^2y/dx^2 \)，是导数的导数。高阶导数用于凹凸性分析和运动学。
Example: if \( f(x) = x^4 \), then \( f'(x) = 4x^3 \), \( f”(x) = 12x^2 \), and \( f”'(x) = 24x \).
例如：若 \( f(x) = x^4 \)，则 \( f'(x) = 4x^3 \)，\( f”(x) = 12x^2 \)，\( f”'(x) = 24x \)。
In kinematics, if \( s(t) \) is displacement, then velocity is \( s'(t) \) and acceleration is \( s”(t) \). This connection is frequently tested in IB applications of differentiation.
在运动学中，若 \( s(t) \) 表示位移，则速度为 \( s'(t) \)，加速度为 \( s”(t) \)。这一关系在IB导数的应用题目中经常出现。
9. Implicit Differentiation | 9. 隐函数求导
Some equations define \( y \) implicitly as a function of \( x \), such as \( x^2 + y^2 = 25 \). Instead of solving for \( y \), we differentiate both sides with respect to \( x \) and use the chain rule on every \( y \)-term.
有些方程将 \( y \) 隐式地定义为 \( x \) 的函数，例如 \( x^2 + y^2 = 25 \)。我们可以不对 \( y \) 求解，而是对等式两边关于 \( x \) 求导，并对每个含 \( y \) 的项使用链式法则。
Differentiate \( x^2 + y^2 = 25 \):
对 \( x^2 + y^2 = 25 \) 两边求导：
2x + 2y × dy/dx = 0
Solving for \( dy/dx \) gives \( dy/dx = -x/y \). At the point \( (3,4) \), the gradient is \( -3/4 \).
解出 \( dy/dx \) 得 \( dy/dx = -x/y \)。在点 \( (3,4) \) 处，切线斜率为 \( -3/4 \)。
Implicit differentiation is especially useful for curves such as circles, ellipses, and related-rate problems in IB Mathematics Analysis and Approaches HL.
隐函数求导在圆、椭圆以及IB数学分析与方法HL中的相关变化率问题中特别有用。
10. Tangent and Normal Lines | 10. 切线与法线
The derivative at a point gives the slope of the tangent line. Once the slope is known, the equation of the line can be written using the point-slope form.
函数在某一点的导数给出该点切线的斜率。已知斜率后，可以用点斜式写出切线方程。
Tangent: y – f(a
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