IB Mathematics: Differentiation Techniques and Applications — IB 数学：微分技术及其应用

一、微分的基本定义：从割线斜率到切线斜率 | The Basic Definition of Differentiation: From Secant Slope to Tangent Slope

微分（Differentiation）是微积分的核心概念之一，它研究的是函数在某一点的瞬时变化率。要理解微分，我们可以从割线（secant line）的斜率出发：给定函数 f(x) 和两点 (a, f(a)) 与 (a+h, f(a+h))，连接这两点的割线斜率为 [f(a+h) – f(a)] / h。当 h 趋近于 0 时，这条割线逐渐趋近于曲线在点 x = a 处的切线，其斜率就是函数在该点的导数。

Differentiation is one of the core concepts of calculus, studying the instantaneous rate of change of a function at a given point. To understand differentiation, we begin with the slope of a secant line: given a function f(x) and two points (a, f(a)) and (a+h, f(a+h)), the secant slope connecting them is [f(a+h) – f(a)] / h. As h approaches 0, this secant line increasingly approaches the tangent line to the curve at x = a, and its slope becomes the derivative of the function at that point.

这个极限过程的数学表达称为”第一原理”（First Principle）：f'(a) = lim(h→0) [f(a+h) – f(a)] / h。IB 数学 AA（Analysis and Approaches）课程要求学生在考试中能够使用第一原理推导基本函数的导数，这是理解后续所有微分法则的基础。例如，通过第一原理可以证明 f(x) = x^n 的导数为 nx^(n-1)。

The mathematical expression of this limiting process is known as the “First Principle”: f'(a) = lim(h→0) [f(a+h) – f(a)] / h. The IB Mathematics AA (Analysis and Approaches) syllabus requires students to derive the derivatives of basic functions from first principles in examinations – this forms the foundation for understanding all subsequent differentiation rules. For example, the first principle can be used to prove that the derivative of f(x) = x^n is nx^(n-1).

二、标准导数公式表：从幂函数到三角函数的求导结果 | Standard Derivative Formula Table: From Power Functions to Trigonometric Derivatives

在掌握第一原理之后，IB 课程要求学生熟记以下标准导数公式。这些结果是反复使用第一原理推导后总结出的规律，学生应能够在不查表的情况下直接使用：

After mastering the first principle, the IB syllabus requires students to memorise the following standard derivative formulae. These results are the summarised patterns derived from repeated application of the first principle, and students should be able to apply them directly without consulting a reference table:

常数函数：d/dx [c] = 0（c 为常数）。这是因为常数函数的图像是一条水平线，斜率为零。幂函数：d/dx [x^n] = nx^(n-1)，这是微积分中最常用的公式，适用于任何实数指数 n。指数函数：d/dx [e^x] = e^x，自然指数函数是其自身的导数，这是数学中最优美的性质之一。对数函数：d/dx [ln x] = 1/x（x > 0）。三角函数：d/dx [sin x] = cos x，d/dx [cos x] = -sin x，d/dx [tan x] = sec^2 x。

Constant function: d/dx [c] = 0 (where c is a constant). This is because the graph of a constant function is a horizontal line with zero slope. Power function: d/dx [x^n] = nx^(n-1), the most frequently used formula in calculus, applicable for any real exponent n. Exponential function: d/dx [e^x] = e^x – the natural exponential function is its own derivative, one of the most elegant properties in mathematics. Logarithmic function: d/dx [ln x] = 1/x (for x > 0). Trigonometric functions: d/dx [sin x] = cos x, d/dx [cos x] = -sin x, d/dx [tan x] = sec^2 x.

在 IB 考试中，这些基本公式是解决更复杂求导问题的起点。一个常见的陷阱是混淆 sin x 和 cos x 的导数符号：sin 的导数是正的 cos，而 cos 的导数是负的 sin。许多学生在对 cos 求导时忘记负号，导致连锁错误。

In IB examinations, these basic formulae are the starting point for solving more complex differentiation problems. A common pitfall is confusing the signs of the derivatives of sin x and cos x: the derivative of sin is positive cos, while the derivative of cos is negative sin. Many students forget the negative sign when differentiating cos, leading to cascading errors.

三、导数的加减法则与常数倍法则：组合函数求导的基础 | The Sum, Difference, and Constant Multiple Rules: The Foundation of Composite Function Differentiation

当函数由多个基本函数的和或差构成时，微分可以逐项进行。具体来说：若 f(x) = u(x) + v(x)，则 f'(x) = u'(x) + v'(x)。同样，若 f(x) = u(x) – v(x)，则 f'(x) = u'(x) – v'(x)。常数倍法则指出：若 f(x) = c * g(x)（c 为常数），则 f'(x) = c * g'(x)。

When a function is composed of the sum or difference of multiple basic functions, differentiation can be performed term by term. Specifically: if f(x) = u(x) + v(x), then f'(x) = u'(x) + v'(x). Similarly, if f(x) = u(x) – v(x), then f'(x) = u'(x) – v'(x). The constant multiple rule states: if f(x) = c * g(x) (where c is a constant), then f'(x) = c * g'(x).

这三条法则的组合使得任何多项式函数都能被轻松求导。例如，对于 f(x) = 3x^4 – 5x^2 + 2x – 7，逐项求导得到 f'(x) = 12x^3 – 10x + 2。IB 考试中常见的要求是首先将函数整理为标准多项式形式，然后逐项求导，这个过程虽然简单但容易因疏忽而出错，尤其是处理负系数时。

The combination of these three rules allows any polynomial function to be differentiated with ease. For example, for f(x) = 3x^4 – 5x^2 + 2x – 7, differentiating term by term yields f'(x) = 12x^3 – 10x + 2. A common requirement in IB examinations is to first rearrange a function into standard polynomial form and then differentiate term by term – a process that is straightforward but prone to careless errors, especially when handling negative coefficients.

四、乘法法则：两个函数乘积的求导技巧 | The Product Rule: Techniques for Differentiating the Product of Two Functions

当需要求两个函数乘积的导数时，不能简单地对每个函数分别求导再相乘。从第一原理出发可以推导出乘法法则（Product Rule）：若 f(x) = u(x) * v(x)，则 f'(x) = u'(x)v(x) + u(x)v'(x)。这个公式的记忆口诀是”先导第一个乘第二个，加第一个乘第二个的导数”。

When differentiating the product of two functions, one cannot simply differentiate each function separately and multiply the results. The Product Rule can be derived from first principles: if f(x) = u(x) * v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). The mnemonic for this formula is “the derivative of the first times the second, plus the first times the derivative of the second.”

在实际应用中，选择哪个函数作为 u(x) 和 v(x) 并不影响最终结果（乘法交换律保证了对称性），但合理的拆分可以简化计算。例如，对于 f(x) = x^2 * ln x，令 u(x) = x^2 和 v(x) = ln x，则 u'(x) = 2x，v'(x) = 1/x，代入乘法法则得 f'(x) = 2x * ln x + x^2 * (1/x) = 2x ln x + x，可进一步因式分解为 x(2 ln x + 1)。

In practice, choosing which function to designate as u(x) and v(x) does not affect the final result (commutativity of multiplication guarantees symmetry), but a sensible split can simplify the computation. For example, for f(x) = x^2 * ln x, setting u(x) = x^2 and v(x) = ln x gives u'(x) = 2x and v'(x) = 1/x. Substituting into the Product Rule yields f'(x) = 2x * ln x + x^2 * (1/x) = 2x ln x + x, which can be further factorised as x(2 ln x + 1).

乘法法则在 IB 考试 Paper 1（不允许使用计算器）中是一个重点考察内容。考试通常涉及多项式与三角函数的乘积（如 x * sin x）或指数函数与多项式的乘积（如 e^x * x^2）。关键是在代入前清晰地写出 u, v, u’, v’，避免漏项或符号错误。

The Product Rule is a heavily tested topic in IB Paper 1 (non-calculator). Examination questions typically involve products of polynomials and trigonometric functions (such as x * sin x) or products of exponential and polynomial functions (such as e^x * x^2). The key is to write out u, v, u’, v’ clearly before substitution to avoid missing terms or sign errors.

五、除法法则：分式函数的求导方法及其与乘法法则的联系 | The Quotient Rule: Methods for Differentiating Rational Functions and Its Connection to the Product Rule

除法法则（Quotient Rule）用于求两个函数之商的导数。若 f(x) = u(x) / v(x)，其中 v(x) 不等于 0，则 f'(x) = [u'(x)v(x) – u(x)v'(x)] / [v(x)]^2。这个公式的记忆口诀是”低导高不导，减去高导低不导，除以低方的平方”，其中”低”指分母 v(x)，”高”指分子 u(x)。

The Quotient Rule is used to differentiate the quotient of two functions. If f(x) = u(x) / v(x), where v(x) is not equal to 0, then f'(x) = [u'(x)v(x) – u(x)v'(x)] / [v(x)]^2. The mnemonic for this formula is “low-d-high minus high-d-low over low-squared,” where “low” refers to the denominator v(x) and “high” refers to the numerator u(x).

需要注意的是，除法法则中分子的减法是 u’v – uv’，而非 uv’ – u’v。这个顺序至关重要 – 搞反符号会导致整个答案错误。一个有效的记忆技巧是意识到分子中的第一项 u’v 从乘法法则中继承而来（因为除法本质上可以看作 u * (1/v)，然后用乘法法则展开），第二项减去 uv’ 来自于对 (1/v) 的求导。

It is important to note that the subtraction in the numerator of the Quotient Rule is u’v – uv’, not uv’ – u’v. The order is critical – reversing the signs leads to an entirely incorrect answer. A useful memorisation technique is to recognise that the first term u’v in the numerator is inherited from the Product Rule (since division can essentially be viewed as u * (1/v) and then expanded using the Product Rule), while the second term minus uv’ comes from differentiating (1/v).

对于某些分式函数，将分母提取为负指数然后用乘法法则或链式法则求导可能比直接用除法法则更简洁。例如，f(x) = (x^2 + 1) / x^3 可以重写为 f(x) = x^(-1) + x^(-3)，然后直接用幂函数的导数公式，避免了除法法则的复杂分子运算。

For certain rational functions, rewriting the denominator with a negative exponent and then applying the Product Rule or Chain Rule may be more concise than using the Quotient Rule directly. For example, f(x) = (x^2 + 1) / x^3 can be rewritten as f(x) = x^(-1) + x^(-3), and then differentiated directly using the power function derivative formula, avoiding the complex numerator computation of the Quotient Rule.

六、链式法则：复合函数求导的核心技术 | The Chain Rule: The Core Technique for Differentiating Composite Functions

链式法则（Chain Rule）是微分学中最重要的工具之一，用于求复合函数的导数。若 y 是 u 的函数，而 u 是 x 的函数，即 y = f(g(x))，则 dy/dx = (dy/du) * (du/dx)，或者写作 f'(g(x)) * g'(x)。这个公式的直观理解是：x 的微小变化通过中间变量 u 逐层传递，最终影响 y。

The Chain Rule is one of the most important tools in differential calculus, used to differentiate composite functions. If y is a function of u, and u is a function of x, i.e., y = f(g(x)), then dy/dx = (dy/du) * (du/dx), or written as f'(g(x)) * g'(x). The intuitive understanding of this formula is that a small change in x propagates through the intermediate variable u layer by layer, ultimately affecting y.

在 IB 考试中，链式法则常与三角函数、指数函数和对数函数组合出现。典型的题目如求 f(x) = sin(2x^3 + 1) 的导数：设外层函数为 sin u，内层函数 u = 2x^3 + 1，则外导数为 cos u，内导数为 6x^2，相乘得 f'(x) = cos(2x^3 + 1) * 6x^2。另一常见题型是 f(x) = e^(kx)，导数为 k * e^(kx)。

In IB examinations, the Chain Rule frequently appears in combination with trigonometric, exponential, and logarithmic functions. A typical problem would be finding the derivative of f(x) = sin(2x^3 + 1): set the outer function as sin u and the inner function as u = 2x^3 + 1, then the outer derivative is cos u, the inner derivative is 6x^2, and multiplying gives f'(x) = cos(2x^3 + 1) * 6x^2. Another common pattern is f(x) = e^(kx), whose derivative is k * e^(kx).

链式法则可以扩展到多层嵌套：对于 y = f(g(h(x)))，导数为 f'(g(h(x))) * g'(h(x)) * h'(x)。处理多层复合函数时，建议从最外层开始逐层向内求导，每一步乘以该层内部函数的导数，这样可以系统性地避免遗漏。

The Chain Rule can be extended to multiple layers of nesting: for y = f(g(h(x))), the derivative is f'(g(h(x))) * g'(h(x)) * h'(x). When handling multiply-nested composite functions, it is advisable to differentiate from the outermost layer inward, multiplying at each step by the derivative of that layer’s inner function – this systematic approach helps avoid omissions.

七、隐函数微分：处理无法显式解出 y 的方程 | Implicit Differentiation: Handling Equations Where y Cannot Be Explicitly Solved

并非所有函数都能方便地写成 y = f(x) 的显式形式。有些方程，如 x^2 + y^2 = 25（圆的方程）或 x^3 + y^3 = 6xy（笛卡尔叶形线），无法将 y 单独解出。隐函数微分（Implicit Differentiation）提供了直接对这类方程求导的方法：对方程两边同时关于 x 求导，将 y 视为 x 的函数，每次对含 y 的项求导后乘以 dy/dx。

Not all functions can be conveniently written in the explicit form y = f(x). Some equations, such as x^2 + y^2 = 25 (the equation of a circle) or x^3 + y^3 = 6xy (the Folium of Descartes), cannot be solved for y alone. Implicit differentiation provides a method for directly differentiating such equations: differentiate both sides of the equation with respect to x, treating y as a function of x, and multiplying by dy/dx after differentiating each term containing y.

以 x^2 + y^2 = 25 为例：对 x^2 求导得 2x；对 y^2 求导（将 y 视为 x 的函数）得 2y * dy/dx；常数 25 的导数为 0。因此 2x + 2y(dy/dx) = 0，解得 dy/dx = -x/y。这个结果可以直接用来求圆上任意点的切线斜率，无需将 y 用 x 表示。

Take x^2 + y^2 = 25 as an example: differentiating x^2 gives 2x; differentiating y^2 (treating y as a function of x) gives 2y * dy/dx; the derivative of the constant 25 is 0. Therefore, 2x + 2y(dy/dx) = 0, yielding dy/dx = -x/y. This result can be used directly to find the slope of the tangent at any point on the circle, without needing to express y in terms of x.

隐函数微分在 IB 数学 HL 级别是必考内容。除了基本圆的切线问题外，还经常考察切线方程和法线方程的求法，以及二阶隐函数求导。二阶隐函数求导的常见方法是先求出一阶导数 dy/dx，然后对该表达式再次求导，注意 dy/dx 本身也是 x 和 y 的函数。

Implicit differentiation is required content at the IB Mathematics HL level. Beyond basic circle tangent problems, examinations frequently test the determination of tangent and normal line equations, as well as second-order implicit differentiation. The common approach to second-order implicit differentiation is to first find the first derivative dy/dx, then differentiate that expression again, noting that dy/dx itself is also a function of x and y.

八、对数微分法：处理复杂指数和乘积的强力工具 | Logarithmic Differentiation: A Powerful Tool for Handling Complex Exponents and Products

当函数涉及变量同时出现在底数和指数位置时（如 f(x) = x^x），或者函数由多个因式的乘积构成时（如 f(x) = (x^2+1)(x-1)^3 / sqrt(x+2)），对数微分法（Logarithmic Differentiation）提供了一条高效的求导路径。基本思路是先对等式两边取自然对数，利用对数性质将指数拉下来或将乘积转化为和的形式，然后使用隐函数微分法。

When a function involves a variable appearing in both the base and the exponent (such as f(x) = x^x), or when a function is composed of the product of multiple factors (such as f(x) = (x^2+1)(x-1)^3 / sqrt(x+2)), logarithmic differentiation provides an efficient differentiation pathway. The basic approach is to take the natural logarithm of both sides of the equation, use logarithmic properties to bring exponents down or convert products into sums, and then apply implicit differentiation.

对于 f(x) = x^x，无法直接使用幂法则（因为指数不是常数）或指数法则（因为底数不是常数）。取自然对数：ln f(x) = x ln x。两边求导：(1/f(x)) * f'(x) = ln x + x * (1/x) = ln x + 1。因此 f'(x) = f(x)(ln x + 1) = x^x (ln x + 1)。

For f(x) = x^x, neither the power rule (because the exponent is not a constant) nor the exponential rule (because the base is not a constant) can be applied directly. Taking the natural logarithm: ln f(x) = x ln x. Differentiating both sides: (1/f(x)) * f'(x) = ln x + x * (1/x) = ln x + 1. Therefore, f'(x) = f(x)(ln x + 1) = x^x (ln x + 1).

IB 数学 AA HL 的考试中可能出现结合对数微分和链式法则的复合题型，例如求 f(x) = (sin x)^(cos x) 的导数。这类题目测试学生能否正确识别何时应该使用对数微分，以及能否在正确的时机调用隐函数微分的技巧。

IB Mathematics AA HL examinations may feature composite problems combining logarithmic differentiation with the Chain Rule, such as finding the derivative of f(x) = (sin x)^(cos x). Such questions test students’ ability to correctly identify when logarithmic differentiation should be used and to deploy implicit differentiation techniques at the right moment.

九、高阶导数及其在运动学和曲线分析中的应用 | Higher-Order Derivatives and Their Applications in Kinematics and Curve Analysis

一阶导数 f'(x) 表示函数的变化率。对该变化率再次求导就得到二阶导数 f”(x)，它表示变化率的变化率 – 即加速度的概念。继续求导可以得到三阶、四阶乃至 n 阶导数。高阶导数在物理学和曲线分析中有广泛应用。

The first derivative f'(x) represents the rate of change of a function. Differentiating this rate of change once more yields the second derivative f”(x), which represents the rate of change of the rate of change – the concept of acceleration. Continuing to differentiate yields third-order, fourth-order, and even nth-order derivatives. Higher-order derivatives have wide-ranging applications in physics and curve analysis.

在运动学（Kinematics）中，若位移函数为 s(t)，则速度 v(t) = s'(t) 是一阶导数，加速度 a(t) = v'(t) = s”(t) 是二阶导数。IB 数学考试中常见的运动学问题包括：已知位移函数求速度和加速度、已知加速度函数求速度（通过积分）、判断粒子何时改变运动方向（v(t) = 0）、以及确定最大位移（v(t) = 0 且 a(t) < 0）。

In kinematics, if the displacement function is s(t), then velocity v(t) = s'(t) is the first derivative, and acceleration a(t) = v'(t) = s”(t) is the second derivative. Common kinematics problems in IB Mathematics examinations include: finding velocity and acceleration from a given displacement function, finding velocity from a given acceleration function (via integration), determining when a particle changes direction of motion (v(t) = 0), and identifying maximum displacement (v(t) = 0 and a(t) < 0).

在曲线分析中，二阶导数用于判断函数的凹凸性（concavity）和寻找拐点（inflection points）。当 f”(x) > 0 时，曲线在该区间内是向上凹的（concave up）；当 f”(x) < 0 时，曲线向下凹（concave down）。拐点出现在 f''(x) 改变符号的位置，此时曲线的凹凸性发生转变。

In curve analysis, the second derivative is used to determine the concavity of a function and to locate inflection points. When f”(x) > 0, the curve is concave up on that interval; when f”(x) < 0, the curve is concave down. Inflection points occur where f''(x) changes sign, at which point the concavity of the curve transitions.

十、驻点分析：用一阶和二阶导数寻找极值点 | Stationary Point Analysis: Using First and Second Derivatives to Find Extreme Points

驻点（Stationary Point）是函数导数等于零的点，即 f'(x) = 0 的位置。在这些点上，函数的切线是水平的，函数的瞬时变化率为零。驻点可以分为三类：极大值点（Maximum Point）、极小值点（Minimum Point）和水平拐点（Point of Inflection with Horizontal Tangent）。

A stationary point is a point where the derivative of a function equals zero, i.e., where f'(x) = 0. At these points, the tangent to the function is horizontal and the instantaneous rate of change is zero. Stationary points can be classified into three types: maximum points, minimum points, and points of inflection with horizontal tangents.

分类驻点的标准方法是二阶导数判别法（Second Derivative Test）：若 f'(a) = 0 且 f”(a) < 0，则该点为极大值点（局部最大值）；若 f'(a) = 0 且 f''(a) > 0，则为极小值点（局部最小值）；若 f”(a) = 0，二阶导数判别法失效，需要进一步检查 f” 在该点附近的符号变化，或使用一阶导数符号判别法。

The standard method for classifying stationary points is the Second Derivative Test: if f'(a) = 0 and f”(a) < 0, the point is a local maximum; if f'(a) = 0 and f''(a) > 0, it is a local minimum; if f”(a) = 0, the Second Derivative Test is inconclusive, and one must further examine the sign change of f” near that point, or use the First Derivative Sign Test.

IB 考试中典型的优化问题（Optimization Problem）要求学生将实际问题建模为数学函数，然后求导找极值。例如，给定一定长度的围栏材料，求可以围出的最大矩形面积；或者给定一个盒子的体积，求使其表面积最小的尺寸。这类题目综合考察建模、求导和驻点分类的能力。

Typical optimisation problems in IB examinations require students to model a real-world scenario as a mathematical function and then differentiate to find extreme values. For example, given a fixed length of fencing material, find the maximum rectangular area that can be enclosed; or given the volume of a box, find the dimensions that minimise its surface area. Such problems comprehensively test modelling, differentiation, and stationary point classification skills.

十一、切线方程与法线方程：几何意义的精确表达 | Tangent and Normal Equations: The Precise Geometric Expression of Differentiation

导数 f'(a) 的几何意义是函数 y = f(x) 在点 (a, f(a)) 处的切线斜率。根据点斜式方程，切线方程为 y – f(a) = f'(a)(x – a)。法线是过同一交点且垂直于切线的直线，其斜率为 -1/f'(a)（前提是 f'(a) 不等于 0），因此法线方程为 y – f(a) = -1/f'(a) * (x – a)。

The geometric meaning of the derivative f'(a) is the slope of the tangent line to the function y = f(x) at the point (a, f(a)). Using the point-slope form, the tangent equation is y – f(a) = f'(a)(x – a). The normal line passes through the same point and is perpendicular to the tangent, with a slope of -1/f'(a) (provided f'(a) is not equal to 0), so the normal equation is y – f(a) = -1/f'(a) * (x – a).

当 f'(a) = 0 时，切线是水平的（平行于 x 轴），法线是垂直的（平行于 y 轴），此时法线方程简化为 x = a。反之，当切线垂直时（函数在该点不可导），法线是水平的。IB 考试中经常要求学生写出给定点处的切线方程和法线方程，这要求准确计算 f(a) 和 f'(a) 并代入公式。

When f'(a) = 0, the tangent is horizontal (parallel to the x-axis) and the normal is vertical (parallel to the y-axis), in which case the normal equation simplifies to x = a. Conversely, when the tangent is vertical (the function is not differentiable at that point), the normal is horizontal. IB examinations frequently require students to write the tangent and normal equations at a given point, which demands accurate computation of f(a) and f'(a) and correct substitution into the formula.

十二、微分在相关变化率问题中的实际应用 | Practical Applications of Differentiation in Related Rates Problems

相关变化率（Related Rates）问题是微积分在现实世界中最直接的应用之一。当一个量随时间变化时，与之相关的其他量也会发生变化。通过微分，我们可以建立这些变化率之间的关系。典型例题包括：一个球形气球被充气，已知体积的增长率，求半径的增长率；或者一架梯子靠墙滑下，已知梯子底端远离墙壁的速度，求梯子顶端下降的速度。

Related rates problems are among the most direct applications of calculus in the real world. When one quantity changes with time, other related quantities also change. Through differentiation, we can establish relationships between these rates of change. Typical examples include: a spherical balloon is being inflated – given the rate of increase of volume, find the rate of increase of the radius; or a ladder sliding down a wall – given the speed at which the foot of the ladder moves away from the wall, find the speed at which the top of the ladder descends.

解决相关变化率问题的标准流程为：(1) 画图并标注变量；(2) 写出连接各变量的方程（通常来自几何关系）；(3) 对方程两边关于时间 t 求导（使用链式法则）；(4) 代入已知值求解未知变化率。注意在代入数值之前必须先完成求导步骤 – 这是 IB 评分标准中的关键要求。

The standard procedure for solving related rates problems is: (1) draw a diagram and label variables; (2) write an equation connecting the variables (typically from a geometric relationship); (3) differentiate both sides of the equation with respect to time t (using the Chain Rule); (4) substitute known values to solve for the unknown rate of change. Note that the differentiation step must be completed before substituting numerical values – this is a key requirement in the IB marking scheme.

Summary | 总结

微分学构成了 IB 数学课程中最核心的板块之一，从第一原理的定义出发，延伸覆盖了标准导数公式、加减乘除复合五种基本求导法则、隐函数微分、对数微分、高阶导数、驻点分析、切线与法线以及相关变化率等丰富主题。掌握这些技术需要系统的公式记忆和大量的练习，尤其是乘法法则、除法法则和链式法则的组合使用。微分不仅是数学工具箱的核心组件，更是物理、工程、经济和数据科学中分析和优化问题的基石。

Differential calculus constitutes one of the most central components of the IB Mathematics curriculum, covering the journey from the first-principles definition through standard derivative formulae, the five fundamental differentiation rules (sum, difference, product, quotient, and chain), implicit differentiation, logarithmic differentiation, higher-order derivatives, stationary point analysis, tangents and normals, and related rates. Mastery of these techniques requires systematic formula memorisation and extensive practice, particularly in the combined use of the Product Rule, Quotient Rule, and Chain Rule. Differentiation is not only a core component of the mathematical toolkit but also the cornerstone for analysing and optimising problems in physics, engineering, economics, and data science.

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