📚 d9U6T5 Mastering Differentiation: Key Concepts and Applications | d9U6T5 微分核心精讲：概念与应用
Differentiation forms the backbone of calculus and is a central topic in Mathematics for the International Student. In this d9U6T5 revision guide, we unpack the essential techniques of differentiation, from first principles to advanced applications including optimisation and related rates. Mastery of these skills is vital for success in both internal assessments and final examinations.
微分是微积分的基石，也是国际学生数学课程中的核心主题。在这份 d9U6T5 复习指南中，我们将系统梳理微分的核心技巧，从第一原理到高阶应用，如最优化与相关变化率。扎实掌握这些技能对于校内外考试都至关重要。
1. The Definition of the Derivative | 导数的定义
The derivative of a function f(x) at a point x = a is defined as the limit of the difference quotient: f'(a) = limh→0 [f(a+h) − f(a)] / h. Geometrically, this represents the slope of the tangent line to the curve y = f(x) at that point. It is the instantaneous rate of change of the function with respect to x.
函数 f(x) 在 x = a 处的导数定义为差商的极限：f'(a) = limh→0 [f(a+h) − f(a)] / h。从几何上看，它表示曲线 y = f(x) 在该点处切线的斜率，也是函数关于 x 的瞬时变化率。
The notation dy/dx, f'(x), and y’ are all used interchangeably. When the limit exists, we say f is differentiable at that point. Differentiability implies continuity, but the converse is not always true.
记号 dy/dx、f'(x) 和 y’ 可互换使用。极限存在时，我们称 f 在该点可导。可导必连续，但连续不一定可导。
2. Power, Sum and Constant Multiple Rules | 幂法则、和法则与常数倍法则
For any real constant n, the derivative of xn is d/dx (xn) = n xn−1. The constant multiple rule states that d/dx [c·f(x)] = c·f'(x), and the sum rule gives d/dx [f(x) ± g(x)] = f'(x) ± g'(x). These basic rules allow us to differentiate any polynomial.
对于任意实数常数 n，xn 的导数为 d/dx (xn) = n xn−1。常数倍法则指出 d/dx [c·f(x)] = c·f'(x)，和法则给出 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。运用这些基本法则可求任意多项式的导数。
d/dx (3x4 − 5x2 + 2x − 7) = 12x3 − 10x + 2
d/dx (3x4 − 5x2 + 2x − 7) = 12x3 − 10x + 2
Rewriting terms with negative or fractional exponents extends the power rule. For instance, √x = x½ has derivative ½ x−½ = 1/(2√x), and 1/x2 = x−2 has derivative −2 x−3.
将各项改写为负指数或分数指数可扩展幂法则。例如 √x = x½ 的导数为 ½ x−½ = 1/(2√x)，1/x2 = x−2 的导数为 −2 x−3。
3. Product and Quotient Rules | 积法则与商法则
When two functions are multiplied, the product rule is used: if y = u(x)v(x), then dy/dx = u’v + uv’. The quotient rule handles division: if y = u(x)/v(x), then dy/dx = (u’v − uv’) / v2. Memorisation of these patterns is essential.
两函数相乘时使用积法则：若 y = u(x)v(x)，则 dy/dx = u’v + uv’。商法则处理除法：若 y = u(x)/v(x)，则 dy/dx = (u’v − uv’) / v2。牢记这些模式非常重要。
d/dx (x2 sin x) = 2x sin x + x2 cos x
d/dx (x2 sin x) = 2x sin x + x2 cos x
A common mistake is to differentiate u and v separately and multiply or divide the results; this is incorrect. Always apply the full rule and simplify afterwards. For quotients, rewriting as a product with a negative exponent can sometimes be an effective alternative.
常见错误是分别对 u 和 v 求导再相乘或相除，这是错误的。务必使用完整法则，随后化简。对于商，也可以改写为带负指数的积形式，有时是有效的替代方法。
4. The Chain Rule | 链式法则
The chain rule is used to differentiate composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). In Leibniz notation, if y = f(u) and u = g(x), then dy/dx = (dy/du)·(du/dx). It is one of the most powerful differentiation tools.
链式法则用于复合函数的求导。若 y = f(g(x))，则 dy/dx = f'(g(x)) · g'(x)。用莱布尼茨记号，若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du)·(du/dx)。这是最强大的微分工具之一。
To differentiate sin(3x2), let u = 3x2 so y = sin u. Then dy/du = cos u, du/dx = 6x, giving dy/dx = cos(3x2)·6x. With practice, this can be done mentally, differentiating ‘outside function first, then multiply by derivative of inside’.
对 sin(3x2) 求导，令 u = 3x2，则 y = sin u，dy/du = cos u，du/dx = 6x，故 dy/dx = cos(3x2)·6x。熟练后可心算：“先对外层函数求导，再乘以内层导数”。
5. 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: d/dx ex = ex. For a general base a, d/dx ax = ax ln a. The derivative of the natural logarithm is d/dx ln x = 1/x for x > 0.
自然指数函数 ex 的导数就是其本身：d/dx ex = ex。对于一般底数 a，d/dx ax = ax ln a。自然对数的导数为 d/dx ln x = 1/x（x > 0）。
Combining with the chain rule yields d/dx ef(x) = ef(x) f'(x) and d/dx ln(f(x)) = f'(x)/f(x). For example, the derivative of e2x is 2e2x, and the derivative of ln(5x) is 5/(5x) = 1/x.
结合链式法则可得 d/dx ef(x) = ef(x) f'(x) 以及 d/dx ln(f(x)) = f'(x)/f(x)。例如，e2x 的导数为 2e2x，ln(5x) 的导数为 5/(5x) = 1/x。
6. Derivatives of Trigonometric Functions | 三角函数的导数
The six basic trigonometric derivatives must be memorised:
- d/dx sin x = cos x
- d/dx cos x = −sin x
- d/dx tan x = sec2 x
- d/dx csc x = −csc x cot x
- d/dx sec x = sec x tan x
- d/dx cot x = −csc2 x
六个基本三角函数的导数必须熟记：
- d/dx sin x = cos x
- d/dx cos x = −sin x
- d/dx tan x = sec2 x
- d/dx csc x = −csc x cot x
- d/dx sec x = sec x tan x
- d/dx cot x = −csc2 x
With the chain rule, d/dx sin(ax+b) = a cos(ax+b). A key point is that angles are always in radians when using calculus derivatives; if a problem gives degrees, convert to radians first.
结合链式法则，d/dx sin(ax+b) = a cos(ax+b)。须注意，在微积分中角度始终采用弧度制；若题目给的是度数，须先转换为弧度。
7. Implicit Differentiation | 隐函数求导
When y is defined implicitly as a function of x, we differentiate both sides of the equation with respect to x, treating y as a function and applying the chain rule. Each derivative of a y-term contributes a factor dy/dx.
当 y 由方程隐式定义为 x 的函数时，我们对方程两边关于 x 求导，把 y 视作 x 的函数并运用链式法则。对 y 的每一项求导都会产生一个因子 dy/dx。
For x2 + y2 = 25: 2x + 2y(dy/dx) = 0 ⇒ dy/dx = −x/y
对于 x2 + y2 = 25：2x + 2y(dy/dx) = 0 ⇒ dy/dx = −x/y
Implicit differentiation is particularly useful for finding gradients of curves that are not functions in the usual sense, such as circles, ellipses, or curves defined by complex equations.
隐函数求导对于寻找非寻常函数曲线的梯度尤为有用，如圆、椭圆或由复杂方程定义的曲线。
8. Higher-Order Derivatives | 高阶导数
The second derivative f”(x) or d2y/dx2 is the derivative of the first derivative. It measures the rate of change of the gradient and is crucial for determining concavity and inflection points. Higher-order derivatives are denoted by f(n)(x).
二阶导数 f”(x) 或 d2y/dx2 是一阶导数的导数，它衡量斜率的变化率，对于确定凹凸性和拐点至关重要。更高阶导数记为 f(n)(x)。
For example, if f(x) = x4, then f'(x) = 4x3, f”(x) = 12x2, f”'(x) = 24x, and f(4)(x) = 24. In kinematics, the second derivative of displacement gives acceleration.
例如，若 f(x) = x4，则 f'(x) = 4x3，f”(x) = 12x2，f”'(x) = 24x，f(4)(x) = 24。在运动学中，位移的二阶导数为加速度。
9. Tangents and Normals | 切线与法线
The equation of the tangent line to y = f(x) at x = a is y − f(a) = f'(a)(x − a). The normal line is perpendicular to the tangent, so its gradient is −1/f'(a), provided f'(a) ≠ 0. The normal equation is y − f(a) = [−1/f'(a)](x − a).
曲线 y = f(x) 在 x = a 处的切线方程为 y − f(a) = f'(a)(x − a)。法线与切线垂直，因此其斜率为 −1/f'(a)（假定 f'(a) ≠ 0），法线方程为 y − f(a) = [−1/f'(a)](x − a)。
These geometric applications are frequently combined with earlier differentiation techniques. Always find the y-coordinate first: f(a). Then substitute into the point-gradient form.
这类几何应用常与此前的微分技巧结合考查。务必先求出 y 坐标 f(a)，再代入点斜式方程。
10. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where f'(x) = 0. We classify them using the second derivative test: if f”(a) > 0, a local minimum; if f”(a) < 0, a local maximum; if f''(a) = 0, the test is inconclusive and we use the first derivative sign change.
驻点出现在 f'(x) = 0 处。我们通过二阶导数检验法分类：若 f”(a) > 0，则为局部极小值；若 f”(a) < 0，则为局部极大值；若 f''(a) = 0，则二阶检验法失效，需用一阶导数变号法判断。
Inflection points occur where the concavity changes, i.e., f”(x) changes sign. In curve sketching, combine intercepts, stationary points, asymptotes and concavity to produce an accurate graph.
拐点出现在凹凸性改变处，即 f”(x) 变号。在描绘曲线时，综合截距、驻点、渐近线和凹凸性，可画出准确的图形。
11. Optimisation Problems | 最优化问题
Optimisation involves finding maximum or minimum values of a quantity in a given context. Establish a function for the quantity to be optimised in terms of one variable, differentiate, set f'(x) = 0, and verify the nature of the stationary point using the second derivative test or boundary values.
最优化问题是寻找给定背景下某个量的最大值或最小值。先建立一个变量表示待优化的量，求导，令 f'(x) = 0，再用二阶导数检验法或边界值验证驻点的性质。
Practical steps: draw a diagram, introduce variables, write the constraint equation, express the quantity to be optimised as a function of a single variable, then differentiate and solve. Common models include minimising surface area for a fixed volume.
实操步骤：画出示意图，引入变量，写出约束方程，将待优化量表示为单一变量的函数，然后求导并求解。常见模型如固定体积下使表面积最小。
12. Related Rates | 相关变化率
Related rates problems involve finding the rate at which one quantity changes by relating it to other quantities whose rates of change are known. Differentiate an equation linking the variables with respect to time t using the chain rule.
相关变化率问题通过将待求变化率的量与已知变化率的量建立联系，从而求出其变化率。利用链式法则对方程两边关于时间 t 求导。
For example, if a ladder slides down a wall, the relationship x2 + y2 = L2 gives 2x(dx/dt) + 2y(dy/dt) = 0. Substituting known values yields the unknown rate. Watch for sign conventions and consistent units.
例如，梯子沿墙壁滑下时，关系式 x2 + y2 = L2 求导得 2x(dx/dt) + 2y(dy/dt) = 0。代入已知值即可求出未知变化率。注意正负号约定与单位统一。
Understanding the d9U6T5 suite of differentiation topics provides a robust foundation for calculus. From limit definitions to real-world modelling, each skill reinforces analytical thinking. Regular practice with varied functions, combined with clear logical presentation, is the key to exam success.
掌握 d9U6T5 板块的微分知识为微积分学习奠定了坚实的基础。从极限定义到实际建模，每一项技能都有助于培养分析思维。通过多样化函数练习和清晰的逻辑表达，是考试成功的关键。
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