IB Mathematics AA: Differentiation from First Principles to Optimisation — IB数学分析与方法：从第一性原理到最优化的微分指南

一、微分到底在算什么：变化率与曲线在一点处的斜率 | What Differentiation Measures: Rate of Change and the Gradient of a Curve at a Point

在 IB 数学分析与方法（Analysis and Approaches，简称 AA）课程中，微分（differentiation）是整个微积分（calculus）板块的第一块基石。它的核心问题只有一个：当一个量在连续变化时，它变化的快慢究竟是多少？例如，一辆汽车在高速公路上行驶，速度表上显示的读数并不是它跑完整段路所用的平均速度，而是它在某一个瞬间的瞬时速度。微分要解决的就是这一类”瞬时变化率”的问题。

In the IB Mathematics Analysis and Approaches (AA) course, differentiation is the first cornerstone of the calculus unit. Its core question is simple: when a quantity is changing continuously, exactly how fast is it changing? For example, the reading on a car’s speedometer is not the average speed over the whole journey, but the instantaneous speed at one particular moment. Differentiation is the tool that answers this kind of “instantaneous rate of change” question.

从几何上看，函数 y = f(x) 的图像是一条曲线。曲线上每一个点的斜率都不一样：上升得陡的地方斜率大，平缓的地方斜率小，下降的地方斜率是负数。函数在 x = a 处的导数 f'(a) 的几何意义，就是曲线在点 (a, f(a)) 处切线的斜率。因此，学会微分，等于同时掌握了”代数上的变化率”和”几何上的切线斜率”两套语言，它们是同一个东西的两种说法。

Geometrically, the graph of a function y = f(x) is a curve. The slope is different at every point on that curve: it is large where the curve rises steeply, small where it is flat, and negative where it is falling. The geometric meaning of the derivative f'(a) at x = a is the slope of the tangent line to the curve at the point (a, f(a)). Learning to differentiate therefore means mastering two interchangeable languages: “rate of change” in algebra and “tangent slope” in geometry. They describe exactly the same object.

二、极限定义：从第一性原理出发求导 | The Limit Definition: Differentiating from First Principles

IB AA 课程要求学生不仅能熟练地套用公式求导，还要能从第一性原理（first principles）出发，用极限的严格定义推导导数。导数 f'(x) 的定义是：f'(x) = lim[h→0] [f(x+h) − f(x)] / h。这个式子的分子 f(x+h) − f(x) 表示当自变量从 x 增加一个小量 h 时，函数值的变化量；分母 h 是自变量的变化量。两者相除得到的是”平均变化率”，而让 h 趋向于 0 取极限，就把平均变化率收敛成了瞬时变化率。

The IB AA course expects students not only to apply differentiation formulas fluently, but also to derive derivatives from first principles using the rigorous limit definition. The derivative is defined as f'(x) = lim[h→0] [f(x+h) − f(x)] / h. The numerator f(x+h) − f(x) is the change in the function value when the input increases by a small amount h, and the denominator h is the change in the input. Their ratio gives the average rate of change, and taking the limit as h approaches 0 converts that average rate into the instantaneous rate.

举例来说，对 f(x) = x² 求导时，先展开 f(x+h) = (x+h)² = x² + 2xh + h²，代入定义得到 [x² + 2xh + h² − x²] / h = (2xh + h²) / h = 2x + h，再令 h → 0，就得到 f'(x) = 2x。这个过程看似繁琐，却是理解”为什么公式成立”的关键，考试中 Paper 1 的非计算器部分经常直接考察这一推导。

For example, to differentiate f(x) = x², first expand f(x+h) = (x+h)² = x² + 2xh + h². Substituting into the definition gives [x² + 2xh + h² − x²] / h = (2xh + h²) / h = 2x + h, and letting h → 0 yields f'(x) = 2x. This process looks laborious, but it is the key to understanding why the formulas work, and the non-calculator section of Paper 1 frequently asks for exactly this derivation.

三、幂函数法则与基本求导公式：快速求导的”快捷键” | The Power Rule and Basic Differentiation Formulas: The Fast Shortcuts

一旦从第一性原理确认了原理，日常计算就依靠一组求导公式。最重要的一条是幂函数法则（power rule）：若 f(x) = xⁿ，则 f'(x) = n·x^(n−1)。这条规则对任意实数指数 n 都成立，例如 x⁵ 的导数是 5x⁴，x^(1/2)（即 √x）的导数是 (1/2)x^(−1/2)。常数函数的导数是 0，因为常数不随 x 变化；常数倍法则告诉我们，若 g(x) = k·f(x)，则 g'(x) = k·f'(x)，即常数可以”提出去”。

Once the principle is confirmed from first principles, everyday computation relies on a set of differentiation formulas. The most important is the power rule: if f(x) = xⁿ, then f'(x) = n·x^(n−1). This rule holds for any real exponent n. For example, the derivative of x⁵ is 5x⁴, and the derivative of x^(1/2) (that is, √x) is (1/2)x^(−1/2). The derivative of a constant is 0 because a constant does not change with x. The constant multiple rule tells us that if g(x) = k·f(x), then g'(x) = k·f'(x), meaning constants can be “pulled out”.

另外还有和差法则：两个函数之和（或之差）的导数，等于各自导数的和（或之差）。所以多项式可以逐项求导：对 f(x) = 3x⁴ − 5x² + 2x − 7 求导，直接得到 f'(x) = 12x³ − 10x + 2。指数函数与自然对数也各有专用公式：e^x 的导数还是它自己 e^x，a^x 的导数是 a^x·ln a，而 ln x 的导数是 1/x。三角函数方面，sin x 的导数是 cos x，cos x 的导数是 −sin x，tan x 的导数是 sec²x。

There is also the sum and difference rule: the derivative of a sum (or difference) is the sum (or difference) of the individual derivatives. This lets us differentiate polynomials term by term: for f(x) = 3x⁴ − 5x² + 2x − 7 we directly get f'(x) = 12x³ − 10x + 2. Exponential and logarithmic functions have their own formulas: the derivative of e^x is e^x itself, the derivative of a^x is a^x·ln a, and the derivative of ln x is 1/x. For trigonometric functions, the derivative of sin x is cos x, the derivative of cos x is −sin x, and the derivative of tan x is sec²x.

四、乘积法则：两个函数相乘时如何求导 | The Product Rule: Differentiating the Product of Two Functions

当函数是两个因式相乘时，绝不能”分别求导再相乘”。乘积法则（product rule）的正确形式是：若 y = u·v，其中 u 和 v 都是关于 x 的函数，则 dy/dx = u’·v + u·v’。也就是说，先让第一个函数求导、第二个保持不变，再让第二个求导、第一个保持不变，最后把两项加起来。很多学生把乘积的导数误记成 u’·v’，这是最常见的失分点之一。

When a function is the product of two factors, you must never “differentiate each one and multiply”. The correct product rule is: if y = u·v, where u and v are both functions of x, then dy/dx = u’·v + u·v’. In words, differentiate the first function and leave the second alone, then differentiate the second and leave the first alone, and finally add the two terms together. Many students wrongly memorise the derivative of a product as u’·v’, and this is one of the most common marks-losing mistakes.

例如求 y = x²·sin x 的导数。这里 u = x²，v = sin x，于是 u’ = 2x，v’ = cos x。代入公式得到 dy/dx = 2x·sin x + x²·cos x。注意两项不能合并，结果必须原样保留。乘积法则在 IB 考试中出现频率极高，尤其是与链式法则、三角函数或指数函数结合时，判断”该用哪条法则”本身就是考点。

For example, differentiate y = x²·sin x. Here u = x² and v = sin x, so u’ = 2x and v’ = cos x. Substituting into the formula gives dy/dx = 2x·sin x + x²·cos x. Note that the two terms cannot be combined, and the result must be left as it is. The product rule appears extremely often in IB exams, and deciding “which rule to use” is itself a tested skill, especially when it is combined with the chain rule, trigonometric functions, or exponentials.

五、商法则：有理函数与分式的求导 | The Quotient Rule: Differentiating Rational Functions and Fractions

商法则（quotient rule）处理的是两个函数相除的情形。若 y = u / v，则 dy/dx = (u’·v − u·v’) / v²。这条公式的结构是”分子先导乘分母，减掉分子乘分母导，整体除以分母的平方”。与乘积法则相比，商法则多了一个负号和一个分母平方，是学生最容易记错顺序的公式。一个有效的记忆口诀是”低导高减高导低，除以低的平方”。

The quotient rule handles the case where one function is divided by another. If y = u / v, then dy/dx = (u’·v − u·v’) / v². The structure is “derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator, all divided by the denominator squared”. Compared with the product rule, the quotient rule adds a minus sign and a squared denominator, making it the formula whose order students most often mix up. A useful memory trick is “low-d-high minus high-d-low, all over low squared”.

例如求 y = x / (x² + 1) 的导数。取 u = x，v = x² + 1，则 u’ = 1，v’ = 2x。代入得到 dy/dx = [1·(x²+1) − x·2x] / (x²+1)² = (1 − x²) / (x²+1)²。这里分母 (x²+1)² 恒为正，所以导数的符号完全由分子 1 − x² 决定：当 |x| < 1 时导数为正、函数递增，当 |x| > 1 时导数为负、函数递减。这个例子说明导数不仅能求出来，还能用来判断函数的增减区间。

For example, differentiate y = x / (x² + 1). Take u = x and v = x² + 1, so u’ = 1 and v’ = 2x. Substituting gives dy/dx = [1·(x²+1) − x·2x] / (x²+1)² = (1 − x²) / (x²+1)². Since the denominator (x²+1)² is always positive, the sign of the derivative is decided entirely by the numerator 1 − x²: the function is increasing when |x| < 1 and decreasing when |x| > 1. This example shows that a derivative is not just a result to compute; it can also be used to determine where a function is increasing or decreasing.

六、链式法则：复合函数的求导利器 | The Chain Rule: The Power Tool for Composite Functions

链式法则（chain rule）是 IB AA 中应用最广、也最容易和前面几条法则混淆的一条。当 y 是 u 的函数、而 u 又是 x 的函数时，y 对 x 的导数等于 y 对 u 的导数乘以 u 对 x 的导数，即 dy/dx = (dy/du)·(du/dx)。通俗地说就是”由外向内、层层求导再相乘”。它处理的是复合函数，例如 y = sin(3x)、y = e^(x²)、y = (2x+1)⁵ 这类”函数套函数”的结构。

The chain rule is the most widely used rule in IB AA and also the easiest to confuse with the others. When y is a function of u and u is a function of x, the derivative of y with respect to x equals the derivative of y with respect to u multiplied by the derivative of u with respect to x: dy/dx = (dy/du)·(du/dx). Colloquially, “work from outside to inside, differentiate each layer and multiply”. It handles composite functions, such as y = sin(3x), y = e^(x²), or y = (2x+1)⁵, where one function is nested inside another.

以求 y = (2x+1)⁵ 为例。外层函数是 u⁵，内层函数是 u = 2x+1。先对外层求导得 5u⁴，再对内层求导得 du/dx = 2，两者相乘并代回 u，得到 dy/dx = 5(2x+1)⁴·2 = 10(2x+1)⁴。同理，y = sin(3x) 的导数是 cos(3x)·3 = 3cos(3x)，y = e^(x²) 的导数是 e^(x²)·2x。记住”内层导数乘出来”这一关键步骤，就能避免丢掉因子而出错。

Take y = (2x+1)⁵ as an example. The outer function is u⁵ and the inner function is u = 2x+1. Differentiate the outer layer to get 5u⁴, then differentiate the inner layer to get du/dx = 2, multiply the two and substitute u back in to obtain dy/dx = 5(2x+1)⁴·2 = 10(2x+1)⁴. Similarly, the derivative of y = sin(3x) is cos(3x)·3 = 3cos(3x), and the derivative of y = e^(x²) is e^(x²)·2x. Remembering the crucial step of “multiplying by the derivative of the inner function” is what stops you from dropping a factor and going wrong.

七、高阶导数与凹凸性：导数的导数告诉我们什么 | Higher Derivatives and Concavity: What the Derivative of the Derivative Tells Us

对导数再求一次导，就得到二阶导数 f”(x)，记作 d²y/dx²。一阶导数描述函数值的变化率（递增还是递减），而二阶导数描述一阶导数本身的变化率，也就是曲线的弯曲方向，即凹凸性（concavity）。若 f”(x) > 0，曲线在该区间向上凹（convex，形如”碗口朝上”）；若 f”(x) < 0，曲线向下凹（concave，形如”碗口朝下”）。二阶导数等于 0 的点通常是凹凸性改变的地方，称为拐点（point of inflection）。

Differentiating the derivative once more gives the second derivative f”(x), written d²y/dx². The first derivative describes how fast the function value is changing (whether it is increasing or decreasing), while the second derivative describes how fast the first derivative itself is changing, that is, the direction in which the curve bends, known as concavity. If f”(x) > 0, the curve is concave up in that interval (shaped like an upward bowl); if f”(x) < 0, the curve is concave down. Points where the second derivative equals 0 are often places where concavity changes, called points of inflection.

二阶导数还能帮助我们区分极大值与极小值。若某驻点处 f'(x) = 0 且 f”(x) < 0，则该点是局部极大值（曲线向下凹，像山顶）；若 f'(x) = 0 且 f”(x) > 0，则是局部极小值（像谷底）；若 f”(x) = 0，则需要进一步用一阶导数的符号变化来判断，这类情形在考试中常作为陷阱出现。高阶导数可以继续求下去，但在 IB AA 范围内，三阶及以上很少直接考察。

The second derivative also helps distinguish maxima from minima. If at a stationary point f'(x) = 0 and f”(x) < 0, the point is a local maximum (the curve is concave down, like a hilltop); if f'(x) = 0 and f”(x) > 0, it is a local minimum (like a valley floor); if f”(x) = 0, we must fall back on the sign change of the first derivative to decide, a case that frequently appears in exams as a trap. Higher derivatives can be computed further, but within the scope of IB AA, third order and above are rarely tested directly.

八、切线与法线：用导数写出直线方程 | Tangents and Normals: Writing Line Equations from the Derivative

导数最直接的应用之一，是求曲线在某一点处的切线（tangent）和法线（normal）。切线在点 (a, f(a)) 处的斜率就是 f'(a)，因此切线的方程可以直接用点斜式写出：y − f(a) = f'(a)·(x − a)。法线是与切线垂直的直线，其斜率是切线斜率的负倒数，即 −1/f'(a)，所以法线方程是 y − f(a) = −(1/f'(a))·(x − a)。两条直线垂直的判据（斜率乘积为 −1）在这里反复使用。

One of the most direct applications of the derivative is finding the tangent and normal lines to a curve at a point. The slope of the tangent at (a, f(a)) is simply f'(a), so the tangent’s equation can be written at once in point-slope form: y − f(a) = f'(a)·(x − a). The normal is the line perpendicular to the tangent, and its slope is the negative reciprocal of the tangent’s slope, namely −1/f'(a), so the normal’s equation is y − f(a) = −(1/f'(a))·(x − a). The criterion for perpendicular lines (the product of their slopes is −1) is used repeatedly here.

例如，求曲线 y = x³ 在点 (1, 1) 处的切线。先求导 f'(x) = 3x²，在 x = 1 处斜率为 3，切线方程即 y − 1 = 3(x − 1)，化简为 y = 3x − 2。法线的斜率是 −1/3，方程为 y − 1 = −(1/3)(x − 1)。这类题目还会反过来问：已知切线的斜率或某条给定直线与曲线相切，求切点坐标或参数值，本质上都是”令 f'(x) 等于已知斜率”然后解方程。

For example, find the tangent to y = x³ at (1, 1). First differentiate to get f'(x) = 3x², so the slope at x = 1 is 3, and the tangent is y − 1 = 3(x − 1), which simplifies to y = 3x − 2. The normal has slope −1/3 and equation y − 1 = −(1/3)(x − 1). These questions are also asked in reverse: given the slope of a tangent, or a given line tangent to the curve, find the point of contact or a parameter value. In essence they all reduce to “set f'(x) equal to a known slope” and then solve the equation.

九、驻点与最优化问题：把现实问题翻译成求导 | Stationary Points and Optimisation: Turning Real-World Problems into Differentiation

最优化（optimisation）是 IB AA 应用题的重头戏。它的思想是：如果一个实际问题可以写成一个关于单个变量的函数，那么函数的最大值或最小值通常出现在导数为零的驻点（stationary point）处，或出现在定义域的端点处。解题步骤是固定的四步：第一步，根据题意设出自变量（通常是一个长度、价格、数量）；第二步，把需要优化的量写成该自变量的函数；第三步，求导并令 f'(x) = 0 解出驻点；第四步，用二阶导数或端点比较来确认是最大值还是最小值，并代回求出最优值。

Optimisation is a major topic in IB AA application questions. The idea is that if a real-world problem can be written as a function of a single variable, then the maximum or minimum of that function usually occurs at a stationary point (where the derivative is zero) or at an endpoint of the domain. The solution method follows four fixed steps: first, define the variable from the problem statement (usually a length, price, or quantity); second, express the quantity to be optimised as a function of that variable; third, differentiate, set f'(x) = 0, and solve for the stationary point; fourth, use the second derivative or endpoint comparison to confirm whether it is a maximum or minimum, then substitute back to find the optimal value.

经典例题是”表面积固定的盒子如何使体积最大”或”用固定长度的篱笆围出最大面积的矩形”。以围篱笆为例：用 100 米篱笆围一个一边靠墙的矩形，设矩形的宽为 x 米，则长是 100 − 2x 米，面积 A = x(100 − 2x) = 100x − 2x²。求导得 A’ = 100 − 4x，令其为零得 x = 25，此时面积最大，最大面积为 25 × 50 = 1250 平方米。考试中这类题一定要写明”为何是最大值”（如 A” = −4 < 0 或说明端点更小），否则会被扣掉结论分。

A classic example is “maximise the volume of a box with fixed surface area”, or “fence the largest rectangular area with a fixed length of fencing”. Take the fencing problem: fence a rectangle with one side against a wall using 100 metres of fencing. Let the width be x metres, so the length is 100 − 2x metres, and the area is A = x(100 − 2x) = 100x − 2x². Differentiating gives A’ = 100 − 4x; setting this to zero gives x = 25, at which point the area is largest, with maximum area 25 × 50 = 1250 square metres. In exams you must always state why it is a maximum (for instance A” = −4 < 0, or note that the endpoints are smaller), otherwise you will lose the conclusion mark.

十、运动学应用：位置、速度与加速度的导数关系 | Kinematics: The Derivative Relationship Between Position, Velocity and Acceleration

微分在 IB AA 中最常见的应用场景之一是运动学（kinematics），即描述物体沿直线运动时的位置、速度与加速度。设物体在时刻 t 的位置为 s(t)，那么速度 v(t) 就是位置对时间的导数，即 v(t) = s'(t)；加速度 a(t) 是速度对时间的导数，也是位置的二阶导数，即 a(t) = v'(t) = s”(t)。这一组关系把”运动”直接翻译成了”求导”：给定位移函数，一次求导得速度，两次求导得加速度。

One of the most common applications of differentiation in IB AA is kinematics, the description of position, velocity, and acceleration for an object moving along a straight line. If the position of an object at time t is s(t), then its velocity v(t) is the derivative of position with respect to time, v(t) = s'(t), and its acceleration a(t) is the derivative of velocity, which is also the second derivative of position: a(t) = v'(t) = s”(t). This set of relationships translates “motion” directly into “differentiation”: given a displacement function, differentiate once for velocity and twice for acceleration.

考试中的典型问法包括：给出 s(t)，求某一时刻的速度或加速度；求物体静止的时刻，即解方程 v(t) = 0；求物体回到出发点的时间，即解 s(t) = 0；以及判断物体在某区间内是加速还是减速。一个必须分清的概念是位移（displacement）与路程（distance）：位移可正可负，是有方向的净变化；路程则始终非负，是物体实际走过的总长度。当物体来回运动时，总路程需要分段计算，把每段速度改变方向的区间分别求位移绝对值再相加，这是失分率很高的一类题。

Typical exam questions include: given s(t), find the velocity or acceleration at a particular instant; find when the object is at rest, that is, solve v(t) = 0; find when it returns to its starting point, that is, solve s(t) = 0; and decide whether the object is speeding up or slowing down over an interval. One concept that must be kept straight is displacement versus distance: displacement can be positive or negative and is the net change with a direction, while distance is always non-negative and is the total length actually travelled. When an object moves back and forth, the total distance must be computed piece by piece, taking the absolute value of the displacement over each interval where the velocity changes sign and then adding them together; this is a type of question with a very high mark-loss rate.

例如，设 s(t) = t³ − 6t² + 9t（单位为米，t 为秒）。速度 v(t) = s'(t) = 3t² − 12t + 9 = 3(t−1)(t−3)，所以物体在 t = 1 秒和 t = 3 秒时静止。加速度 a(t) = v'(t) = 6t − 12。在 0 到 1 秒之间物体沿正方向运动，1 到 3 秒之间沿负方向运动，因此 0 到 4 秒的总路程要把 [0,1]、[1,3]、[3,4] 三段位移的绝对值分别加起来，而不是简单地算 s(4) − s(0)。这类题把微分、因式分解与分段绝对值综合在一起，是典型的 IB 综合应用。

For example, let s(t) = t³ − 6t² + 9t (in metres, with t in seconds). The velocity is v(t) = s'(t) = 3t² − 12t + 9 = 3(t−1)(t−3), so the object is at rest at t = 1 and t = 3 seconds. The acceleration is a(t) = v'(t) = 6t − 12. Between 0 and 1 second the object moves in the positive direction, and between 1 and 3 seconds it moves in the negative direction, so the total distance from 0 to 4 seconds must be found by adding the absolute values of the displacement over the three intervals [0,1], [1,3], and [3,4], rather than simply computing s(4) − s(0). This kind of problem combines differentiation, factorisation, and piecewise absolute values, making it a typical IB integrated application.

十一、应试技巧：微分题目的高频错误与规避方法 | Exam Technique: Common Differentiation Mistakes and How to Avoid Them

IB 考试中微分相关的失分往往不是”不会做”，而是”会做但做错”。最需要警惕的几类错误包括：第一，混淆乘积法则与商法则的顺序，尤其是商法则分子里的负号；第二，链式法则漏乘内层导数，例如把 sin(3x) 的导数误写成 cos(3x) 而丢掉系数 3；第三，幂函数法则与指数函数法则搞混，把 e^x 的导数误写成 x·e^(x−1)；第四，在求驻点后忘记确认最大值还是最小值，导致结论不完整；第五，求切线时把函数值 f(a) 与导数 f'(a) 的位置搞错。

In IB exams, marks lost on differentiation are often not “I do not know how” but “I knew how and got it wrong”. The most dangerous categories of error are: first, mixing up the order in the product and quotient rules, especially the minus sign in the quotient rule’s numerator; second, forgetting to multiply by the inner derivative in the chain rule, for example writing the derivative of sin(3x) as cos(3x) and dropping the factor 3; third, confusing the power rule with the exponential rule, and wrongly writing the derivative of e^x as x·e^(x−1); fourth, finding a stationary point but forgetting to confirm whether it is a maximum or minimum, leaving the conclusion incomplete; fifth, mixing up the function value f(a) and the derivative f'(a) when finding a tangent.

有效的应对策略是：每一步都问自己”这里用的是哪条法则”，并把公式写在草稿上再代入；链式法则永远把内层函数的导数用方括号标出来单独写一行；对最优化问题，养成”求导、令零、解方程、验证、代回”五步缺一不可的习惯。Paper 1 不允许使用计算器，因此对基础公式的熟练度直接决定得分；平时练习时建议刻意手写完整的求导过程，而不是跳过中间步骤，这样到了考场才能又快又稳。

An effective counter-strategy is to ask yourself at every step “which rule am I using here”, write the formula down on scratch paper before substituting, and always write the inner function’s derivative on its own line in square brackets when using the chain rule. For optimisation problems, form the habit of five non-negotiable steps: differentiate, set to zero, solve, verify, and substitute back. Paper 1 does not allow a calculator, so fluency with the basic formulas directly determines your score. When practising, deliberately write out the full differentiation process rather than skipping intermediate steps, so that in the exam you are both fast and reliable.

Summary | 总结

微分是 IB 数学分析与方法课程的核心工具，它把”变化率”与”切线斜率”这两个概念统一在一起。本文从第一性原理的极限定义出发，梳理了幂函数法则、乘积法则、商法则与链式法则这四条基础求导法则，进而延伸到高阶导数与凹凸性、切线与法线，以及最优化问题，最后归纳了考试中的高频错误与应对策略。掌握微分的关键，是理解每条法则”为什么成立”和”何时使用”，并在大量的手写练习中把它们内化为肌肉记忆。

Differentiation is the core tool of the IB Mathematics Analysis and Approaches course, unifying the two ideas of “rate of change” and “tangent slope”. Starting from the limit definition at first principles, this article has walked through the four fundamental rules (the power rule, product rule, quotient rule, and chain rule), then extended to higher derivatives and concavity, tangents and normals, and optimisation problems, before summarising the most frequent exam mistakes and how to avoid them. The key to mastering differentiation is understanding why each rule holds and when to use it, and internalising them into muscle memory through plenty of handwritten practice.

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