📚 Further Techniques for Differentiation | 求导的进阶技巧
In IB Mathematics, differentiation extends well beyond simple power, product and quotient rules. You must be comfortable with implicit differentiation, parametric equations, logarithmic techniques, higher-order derivatives, inverse functions, related rates, and advanced limit methods. This article walks through each technique with worked examples to build fluency for both analysis and applications.
在 IB 数学中，求导远远不止简单的幂函数、乘积和商法则。你必须熟练掌握隐函数求导、参数方程、对数技巧、高阶导数、反函数、相关变化率以及高级极限方法。本文通过解题示例逐一讲解每个技巧，帮助你在分析和应用中得心应手。
1. Implicit Differentiation | 隐函数求导
(x, y) = 0 and it is difficult or impossible to solve for y explicitly as a function of x.
当 x 与 y 的关系由一个方程 F(x, y) = 0 给出，且难以或无法显式解出 y 为 x 的函数时，我们使用隐函数求导。
We differentiate both sides of the equation with respect to x, treating y as a differentiable function of x and applying the chain rule to any term containing y. For example, the derivative of y³ with respect to x is 3y²·dy/dx.
我们对方程两边关于 x 求导，将 y 视为 x 的可微函数，并对任何包含 y 的项应用链式法则。例如，y³ 对 x 的导数为 3y²·dy/dx。
Consider the circle x² + y² = 25. Differentiating gives 2x + 2y·dy/dx = 0, hence
dy/dx = −x/y.
考虑圆 x² + y² = 25。求导得 2x + 2y·dy/dx = 0，因此 dy/dx = −x/y。
or a more involved implicit equation such as x² + xy + y² = 7, differentiate term by term:
对于更复杂的隐式方程，如 x² + xy + y² = 7，逐项求导：
d/dx (x²) = 2x, d/dx (xy) = x·dy/dx + y (product rule), and d/dx (y²) = 2y·dy/dx.
d/dx (x²) = 2x，d/dx (xy) = x·dy/dx + y（乘积法则），d/dx (y²) = 2y·dy/dx。
+ y + x·dy/dx + 2y·dy/dx = 0. Collecting dy/dx terms yields (x + 2y)·dy/dx = −(2x + y), so
dy/dx = −(2x + y)/(x + 2y).
得到 2x + y + x·dy/dx + 2y·dy/dx = 0。整理含有 dy/dx 的项得 (x + 2y)·dy/dx = −(2x + y)，所以 dy/dx = −(2x + y)/(x + 2y)。
This derivative can then be used to find slopes of tangents or to compute second derivatives by further implicit differentiation.
该导数可以用于求切线斜率，或通过进一步的隐函数求导计算二阶导数。
2. Logarithmic Differentiation | 对数求导
Logarithmic differentiation simplifies the differentiation of functions involving products, quotients or powers by taking the natural logarithm of both sides first.
对数求导通过先对两边取自然对数，简化包含乘积、商或幂的函数的求导。
Given y = f(x), write ln|y| = ln|f(x)| he right-hand side, then differentiate implicitly. Multiply through by y to recover dy/dx.
给定 y = f(x)，写出 ln|y| = ln|f(x)|。利用对数法则展开右边，然后隐式求导。最后两边乘以 y 即可得到 dy/dx。
Example: y = (x²+1)³ √x−1 / (x+2)⁴. Taking logs gives ln y = 3 ln(x²+1) + ½ ln(x−1) − 4 ln(x+2).
示例：y = (x²+1)³ √x−1 / (x+2)⁴。取对数得 ln y = 3 ln(x²+1) + ½ ln(x−1) − 4 ln(x+2)。
Differentiating, we get (1/y) dy/dx = 3·(2x/(x²+1)) + ½·(1/(x−1)) − 4·(1/(x+2)).
求导得 (1/y) dy/dx = 3·(2x/(x²+1)) + ½·(1/(x−1)) − 4·(1/(x+2))。
(x−1)) − 4/(x+2)]. This technique is much more efficient than using product and quotient rules directly.
因此 dy/dx = y [6x/(x²+1) + 1/(2(x−1)) − 4/(x+2)]。这种方法比直接使用乘积和商法则高效得多。
3. Parametric Differentiation – First Derivative | 参数方程求导——一阶导数
by x = f(t), y = g(t), the chain rule gives dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.
当曲线由参数方程 x = f(t), y = g(t) 定义时，由链式法则得 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。
This formula allows us to find the gradient of the curve directly in terms of the parameter t.
该公式使我们能直接通过参数 t 求曲线的梯度。
For example, if x = t² + t and y = 2t − 1, then dx/dt = 2t + 1, dy/dt = 2. Hence
dy/dx = 2/(2t + 1).
例如，若 x = t² + t 且 y = 2t − 1，则 dx/dt = 2t + 1，dy/dt = 2。因此 dy/dx = 2/(2t + 1)。
t = 1: dy/dx = 2/3, and the point is (2, 1). The tangent equation is y − 1 = (2/3)(x − 2).
要求 t = 1 处的切线，代入 t = 1：dy/dx = 2/3，对应点为 (2, 1)。切线方程为 y − 1 = (2/3)(x − 2)。
4. Parametric Second Derivative | 参数方程的二阶导数
The second derivative in
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