📚 Further Differentiation | 进阶微分
Further differentiation extends the basic rules of calculus to a wider class of functions. In the Cambridge A-Level syllabus, this includes exponential, logarithmic, trigonometric and inverse trigonometric functions, as well as techniques for differentiating implicit and parametric equations. Mastering these skills is essential for solving problems in kinematics, optimisation and curve sketching.
进阶微分将微积分的基本法则推广到更广泛的函数类型。在剑桥 A-Level 课程大纲中，这包括指数函数、对数函数、三角函数与反三角函数的微分，以及隐函数和参数方程的微分技巧。掌握这些技能对于解决运动学、优化问题和曲线作图等问题至关重要。
1. Review of Core Rules | 核心法则回顾
Before exploring new functions, recall the three fundamental rules. The product rule states that d/dx (uv) = u(dv/dx) + v(du/dx); the quotient rule gives d/dx (u/v) = (v du/dx − u dv/dx)/v²; and the chain rule states d/dx f(g(x)) = f'(g(x))g'(x).
在学习新函数之前，先回顾三条基本法则。乘积法则指出 d/dx (uv) = u(dv/dx) + v(du/dx)；商法则给出 d/dx (u/v) = (v du/dx − u dv/dx)/v²；链式法则表明 d/dx f(g(x)) = f'(g(x))g'(x)。
The power rule d/dx (xⁿ) = nxⁿ⁻¹ remains the backbone for polynomial differentiation. These three rules combine with the new derivatives in this article to solve nearly every A-Level differentiation problem.
幂法则 d/dx (xⁿ) = nxⁿ⁻¹ 仍然是多项式微分的基础。这三条法则与本文中的新导数相结合，可以解决几乎所有 A-Level 微分问题。
2. Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The natural exponential function is its own derivative: d/dx (eˣ) = eˣ. This remarkable property makes e the most natural base for calculus. More generally, by the chain rule, d/dx (eᵘ) = eᵘ du/dx. For a general positive base a, we have d/dx (aˣ) = aˣ ln a.
自然指数函数的导数等于其自身：d/dx (eˣ) = eˣ。这一非凡性质使 e 成为微积分中最自然的底数。更一般地，根据链式法则，d/dx (eᵘ) = eᵘ du/dx。对于一般的正底数 a，我们有 d/dx (aˣ) = aˣ ln a。
For logarithms, d/dx (ln x) = 1/x, and d/dx (logₐ x) = 1/(x ln a). A particularly useful result combines the chain rule with logarithms: d/dx (ln f(x)) = f'(x)/f(x). This technique, known as logarithmic differentiation, simplifies derivatives of complicated products and powers.
对于对数函数，d/dx (ln x) = 1/x，d/dx (logₐ x) = 1/(x ln a)。一个特别有用的结果是将链式法则与对数结合：d/dx (ln f(x)) = f'(x)/f(x)。这种技巧称为对数微分法，可以简化复杂乘积和幂函数的求导。
Example: find the derivative of y = xˣ. Taking natural logarithms on both sides gives ln y = x ln x. Differentiating implicitly with respect to x:
示例：求 y = xˣ 的导数。两边取自然对数得 ln y = x ln x。关于 x 隐式求导：
(1/y) dy/dx = ln x + 1, hence dy/dx = xˣ(ln x + 1)
3. Trigonometric Functions | 三角函数的微分
The derivatives of the six standard trigonometric functions must be memorised. For sine and cosine: d/dx (sin x) = cos x and d/dx (cos x) = −sin x. For tangent: d/dx (tan x) = sec²x. The reciprocal functions follow: d/dx (sec x) = sec x tan x, d/dx (cosec x) = −cosec x cot x, and d/dx (cot x) = −cosec²x.
六个标准三角函数的导数必须牢记。对于正弦和余弦：d/dx (sin x) = cos x，d/dx (cos x) = −sin x。对于正切：d/dx (tan x) = sec²x。倒数函数如下：d/dx (sec x) = sec x tan x，d/dx (cosec x) = −cosec x cot x，d/dx (cot x) = −cosec²x。
When these functions are composed with another function, apply the chain rule. For example, d/dx (sin(3x²)) = cos(3x²) × 6x = 6x cos(3x²). Likewise, d/dx (tan(4x + 1)) = 4 sec²(4x + 1).
当这些函数与另一个函数复合时，运用链式法则。例如，d/dx (sin(3x²)) = cos(3x²) × 6x = 6x cos(3x²)。类似地，d/dx (tan(4x + 1)) = 4 sec²(4x + 1)。
4. Inverse Trigonometric Functions | 反三角函数的微分
The derivatives of inverse trigonometric functions appear frequently in integration problems and are derived from the chain rule applied to inverse functions. The three key results are:
反三角函数的导数经常出现在积分问题中，可通过链式法则作用于反函数来推导。三个关键结果为：
d/dx (arcsin x) = 1/√(1 − x²), d/dx (arccos x) = −1/√(1 − x²), d/dx (arctan x) = 1/(1 + x²)
Note the domains: arcsin x and arccos x are defined for −1 ≤ x ≤ 1, while arctan x is defined for all real x. By the chain rule, d/dx (arcsin(x/2)) = (1/2) × 1/√(1 − x²/4) = 1/√(4 − x²).
注意定义域：arcsin x 和 arccos x 的定义域为 −1 ≤ x ≤ 1，而 arctan x 对所有实数 x 有定义。由链式法则，d/dx (arcsin(x/2)) = (1/2) × 1/√(1 − x²/4) = 1/√(4 − x²)。
5. Implicit Differentiation | 隐函数微分
When a curve is defined implicitly by an equation such as x² + y² = 25, we differentiate every term with respect to x and use the chain rule for terms involving y: d/dx (y²) = 2y dy/dx. Differentiating the equation gives 2x + 2y dy/dx = 0, so dy/dx = −x/y.
当曲线由隐式方程（如 x² + y² = 25）定义时，我们对每一项关于 x 求导，并对含有 y 的项使用链式法则：d/dx (y²) = 2y dy/dx。对等式求导得 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。
At the point (3, 4) on this circle, the gradient of the tangent is −3/4, so the normal has gradient 4/3. The tangent equation is y − 4 = (−3/4)(x − 3), which simplifies to 3x + 4y = 25.
在该圆上的点 (3, 4) 处，切线斜率为 −3/4，因此法线斜率为 4/3。切线方程为 y − 4 = (−3/4)(x − 3)，化简得 3x + 4y = 25。
Implicit differentiation is especially valuable for curves that cannot be written as y = f(x), such as the folium of Descartes: x³ + y³ = 6xy. Differentiating term by term gives 3x² + 3y² dy/dx = 6y + 6x dy/dx, from which dy/dx = (2y − x²)/(y² − 2x).
隐函数微分对于无法写成 y = f(x) 形式的曲线尤其有价值，例如笛卡尔叶形线：x³ + y³ = 6xy。逐项求导得 3x² + 3y² dy/dx = 6y + 6x dy/dx，从而 dy/dx = (2y − x²)/(y² − 2x)。
6. Parametric Differentiation | 参数方程微分
For curves defined by parametric equations x = f(t) and y = g(t), the chain rule provides dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This formula converts the rates of change with respect to the parameter into the gradient of the curve.
对于由参数方程 x = f(t)、y = g(t) 定义的曲线，链式法则给出 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。该公式将关于参数的变化率转换为曲线的斜率。
Example: for x = t², y = 2t, we have dx/dt = 2t and dy/dt = 2, so dy/dx = 2/(2t) = 1/t. At t = 2, the gradient is 1/2. The equation of the tangent at this point is y − 4 = (1/2)(x − 4).
示例：对于 x = t²，y = 2t，有 dx/dt = 2t，dy/dt = 2，因此 dy/dx = 2/(2t) = 1/t。当 t = 2 时，斜率为 1/2。该点的切线方程为 y − 4 = (1/2)(x − 4)。
When the parameter is time t, dy/dx represents the slope of the trajectory, while dx/dt and dy/dt are the horizontal and vertical components of velocity.
当参数为时间 t 时，dy/dx 表示轨迹的斜率，而 dx/dt 和 dy/dt 分别是速度的水平分量和竖直分量。
7. Tangents, Normals and Connected Rates | 切线、法线与相关变化率
The gradient obtained by differentiation gives the slope of the tangent directly. The normal at the same point has gradient −1/m when m ≠ 0, and is vertical when m = 0. For connected rates of change, the chain rule links multiple rates: dV/dt = dV/dr × dr/dt.
微分求得的斜率直接给出切线的斜率。同一点处法线的斜率为 −1/m（当 m ≠ 0 时）；当 m = 0 时法线为竖直线。对于相关变化率，链式法则将多个变化率联系起来：dV/dt = dV/dr × dr/dt。
Example: a spherical balloon expands so that its radius increases at a constant rate of 2 cm/s. The volume is V = (4/3)πr³, so dV/dr = 4πr². At the instant when r = 5 cm, dV/dt = 4π(25) × 2 = 200π cm³/s, approximately 628 cm³/s.
示例：一个球形气球膨胀，其半径以 2 cm/s 的恒定速率增加。体积为 V = (4/3)πr³，因此 dV/dr = 4πr²。在 r = 5 cm 的时刻，dV/dt = 4π(25) × 2 = 200π cm³/s，约为 628 cm³/s。
Another classic problem: a ladder 5 m long slides down a wall. If x is the distance of the foot from the wall and y is the height of the top, then x² + y² = 25. Differentiating with respect to time t: 2x dx/dt + 2y dy/dt = 0. Given dx/dt and the position, dy/dt can be found immediately.
另一个经典问题：一架 5 米长的梯子沿墙滑下。设 x 为梯脚离墙的距离，y 为顶端高度，则 x² + y² = 25。关于时间 t 求导：2x dx/dt + 2y dy/dt = 0。已知 dx/dt 和位置，即可立即求出 dy/dt。
8. Second Derivatives for Implicit and Parametric Curves | 隐函数与参数曲线的二阶导数
For implicit curves, differentiate the first derivative expression again. From x² + y² = 25 we found dy/dx = −x/y. Using the quotient rule
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