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Differentiating Composite Functions f(ax+b) | 复合函数 f(ax+b) 的求导

2026-08-30 16:58 · 本文部分内容由AI辅助创作

📚 Differentiating Composite Functions f(ax+b) | 复合函数 f(ax+b) 的求导

In IB Mathematics, the chain rule is one of the most essential tools of differentiation. A very common special case appears when the inner function is linear: that is, when you differentiate a composite function of the form f(ax+b). This pattern shows up constantly in exam papers, and mastering it saves time and prevents errors.

在 IB 数学中，链式法则是最基本的求导工具之一。当内层函数为线性函数时——也就是对形如 f(ax+b) 的复合函数求导——会出现一个非常常见的特殊情况。这个规律在考卷中频繁出现，掌握它既能节省时间，又能避免错误。

1. The Chain Rule Reminder | 链式法则回顾

rule states that if y = f(u) and u = g(x), then the derivative of y with respect to x is:

链式法则指出：若 y = f(u)，且 u = g(x)，则 y 关于 x 的导数为：

dy/dx = dy/du × du/dx

x). The derivative is the product of the derivative of the outer function (evaluated at g(x)) and the derivative of the inner function.

用函数记号则常写成 d/dx f(g(x)) = f′(g(x)) g′(x)。导数为外层函数在 g(x) 处的导数与内层函数导数的乘积。

2. The General Pattern | 一般模式

e inner function is linear, g(x) = ax + b, the chain rule simplifies to a very clean rule:

当内层函数是线性函数，即 g(x) = ax + b 时，链式法则可以简化为一个非常干净利落的规则：

d/dx f(ax + b) = a f′(ax + b)

All you do is differentiate the outer function at the linear expression (ax + b) and then multiply by the constant a.

你只需要在线性表达式 (ax + b) 处对外层函数求导，再乘以常数 a 即可。

3. Why the Factor a Appears | 系数 a 的由来

ternatively, apply the chain rule mechanically: g′(x) = a, so the multiplier is exactly a.

要理解系数 a 为何出现，可以比较 f(x) 与 f(ax+b)。当 x 增加一个单位时，f 的自变量变化了 a 个单位而非 1 个单位，因此变化率被放大了 a 倍。或者直接从链式法则推演：g′(x) = a，所以乘数正好为 a。

) = 8x+4. You can verify by expanding: (2x+1)² = 4x² + 4x + 1, whose derivative is 8x + 4. ✓

例如：设 f(x) = x²，考察 f(2x+1)。此时 f′(u) = 2u，所以 f′(2x+1) = 2(2x+1)。再乘以 a = 2 得到 4(2x+1) = 8x+4。你甚至可以展开验证：(2x+1)² = 4x² + 4x + 1，其导数为 8x + 4。✓

4. Polynomial Examples | 多项式实例

Let f(x) = x⁵. Then f(3x − 2) = (3x − 2)⁵. Applying the rule:

设 f(x) = x⁵，则 f(3x − 2) = (3x − 2)⁵。运用上面的规则：

d/dx (3x − 2)⁵ = 5(3x − 2)⁴ × 3 = 15(3x − 2)⁴

For f(x) = x³ with f(4x+1): d/dx (4x+1)³ = 3(4x+1)² × 4 = 12(4x+1)².

对于 f(x) = x³，若 f(4x+1)：d/dx (4x+1)³ = 3(4x+1)² × 4 = 12(4x+1)²。

Notice that the exponent is reduced by one first, exactly as in the ordinary power rule, and then the factor a is appended.

注意：指数先像普通幂函数求导那样减 1，然后乘上因子 a。

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5. Trigonometric Functions | 三角函数

For trigonometric functions the same principle holds. The derivatives of sin, cos, and tan are applied to the whole argument (ax+b), then multiplied by a.

对于三角函数，同样的原理也成立。sin、cos、tan 的导数先作用于整个内层表达式 (ax+b)，再乘以 a。

d/dx sin(2x+1) = 2 cos(2x+1).

d/dx sin(2x+1) = 2 cos(2x+1)。

d/dx cos(5x − π/3) = −5 sin(5x − π/3).

d/dx cos(5x − π/3) = −5 sin(5x − π/3)。

d/dx tan(4x) = 4 sec²(4x).

d/dx tan(4x) = 4 sec²(4x)。

Note that the argument of the trigonometric function is unchanged in the derivative; only the factor in front is added.

注意：三角函数内部的变量表达式在导数中保持不变，只是前面多乘了系数。

6. Exponential and Logarithmic Functions | 指数与对数函数

For the exponential function with base e, the derivative of eax+b is eax+b multiplied by a:

对于以 e 为底的指数函数，eax+b 的导数为 eax+b 乘以 a：

d/dx e3x+2 = 3e3x+2

For a general base, use the rule d/dx bᵘ = bᵘ ln(b) × du/dx. Hence:

对于一般的底数，可使用公式 d/dx bᵘ = bᵘ ln(b) × du/dx。因此：

d/dx 24x−1 = 24x−1 ln 2 × 4 = 4 ln 2 × 24x−1

For logarithms, d/dx ln(ax+b) = a/(ax+b). For example:

对于对数函数，d/dx ln(ax+b) = a/(ax+b)。例如：

d/dx ln(5x−1) = 5/(5x−1)

It is a common error to write 1/(ax+b) without the factor a. Always check the numerator.

一个常见错误是漏写系数 a，错误地写成 1/(ax+b)。务必检查分子。

7. Common Pitfalls | 常见错误