📚 Mixed Exercise 12: Differentiation | 综合练习12：微分
Mixed Exercise 12 in the Edexcel AS Pure Mathematics textbook brings together all the key differentiation techniques covered in Chapter 12. This exercise is designed to test your ability to find derivatives from first principles, compute gradients of curves, determine equations of tangents and normals, analyse stationary points, and apply differentiation to real‑world problems. Working through these mixed questions will help you consolidate understanding, spot common pitfalls, and build the fluency needed for the exam.
Edexcel 纯数学 AS 课本中的综合练习 12 汇集了第 12 章所涵盖的所有核心微分技巧。该练习旨在检验你从第一原理求导、计算曲线梯度、确定切线和法线方程、分析驻点以及将微分应用于实际问题的能力。完成这些混合题目有助于巩固理解、发现常见陷阱，并培养考试所需的一定熟练度。
1. Differentiation from First Principles | 第一原理求导
a simple function like f(x) = x² or f(x) = x³ using the limit definition. The formula is f'(x) = limₕ→₀ [f(x+h) – tting h approach zero. Always write the limit statement at each step to show full working.
许多题目一开始会要求用极限定义对简单函数如 f(x) = x² 或 f(x) = x³ 求导。公式为 f'(x) = limₕ→₀ [f(x+h) – f(x)] / h。你必须仔细展开括号并在让 h 趋于零之前化简代数式。每一步都要写出极限符号，以展示完整的推导过程。
2. Differentiating Polynomials and Power Functions | 多项式和幂函数的求导
/dx (xⁿ) = n xⁿ⁻¹ is essential. For polynomials, differentiate term by term. Remember that √x can be written as x½, and 1/x² as x⁻². Coefficients remain unchanged, and constant terms differentiate to zero. Always present your final answer with positive indices where possible and in simplest form.
幂法则 d/dx (xⁿ) = n xⁿ⁻¹ 是基本的。对于多项式，逐项求导即可。记住 √x 可写成 x½，1/x² 写成 x⁻²。系数保持不变，常数项求导为零。最终答案应尽可能用正指数并表示成最简形式。
3. Finding the Gradient of a Curve at a Point | 求曲线在一点的梯度
To find the gradient at a specific point, differentiate the function and then substitute the x‑coordinate into f'(x). This gives the gradient of the tangent at that point. Watch out for fractional or negative gradients – the sign indicates whether the curve is increasing or decreasing at that instant.
要求某一点的梯度，先对函数求导，然后将 x 坐标代入 f'(x)。所得结果就是该点切线的梯度。注意分数或负梯度 —— 正负号表示曲线在该点是递增还是递减。
4. Equations of Tangents and Normals | 切线与法线方程
After finding the gradient m of the tangent at (x₁, y₁), use y – y₁ = m(x – o get the tangent equation. The normal is perpendicular to the tangent, so its gradient is –1/m (provided m ≠ 0). Substitute into the same straight‑ne formula. Always give the final equation in the requested form, such as ax + by + c = 0.
求出切点 (x₁, y₁) 处切线的梯度 m 后，用 y – y₁ = m(x – x₁) 得到切线方程。法线与切线垂直，因此其梯度为 –1/m（只要 m ≠ 0）。代入同一直线公式即可。最终答案务必写成要求的格式，例如 ax + by + c = 0。
5. Increasing and Decreasing Functions | 递增函数与递减函数
A function is increasing where f'(x) > 0 and decreasing where f' you to find the interval of x for which the function is increasing. Solve the inequality f' by considering critical points and sketching a sign diagram. Don’t include points where f'(x) = 0 unless the question specifies otherwise.
当 f'(x) > 0 时函数递增，当 f'(x) < 0 时递减。综合练习题常要求找出函数递增的 x 区间。通过找出临界点并绘制符号图来解不等式 f'(x) > 0。除非题目特别说明，否则不要包含 f'(x) = 0 的点。
6. Stationary Points, Turning Points and Their Nature | 驻点、拐点及其性质
Stationary points occur where f' their nature, use either the second derivative test or a gradient table. For a maximum, f''(x) < 0; for a minimum, f''(x) > 0. If f''(x) = 0, examine the sign change of f'(x) on either side. Always state the coordinates and the type of stationary point clearly.
驻点出现在 f'(x) = 0 处。要确定其性质，可使用二阶导数判别法或梯度表。极大值时 f''(x) < 0；极小值时 f''(x) > 0。若 f''(x) = 0，则需考察 f'(x) 两侧的符号变化。务必清楚地写出坐标和驻点类型。
7. Using the Second Derivative and Points of Inflection | 二阶导数与 inflection 点的使用
flection occurs where the curve changes from convex to concave (or vice versa). Although f''(x) = 0 is necessary, it is not sufficient – you must confirm a change in the sign of f''(x). Mixed exercises will test your understanding of this subtle distinction.
二阶导数反映梯度的变化率。当曲线从凸变为凹（或相反）时，会出现 inflection 点。虽然需要 f''(x) = 0，但这还不够 —— 你必须确认 f''(x) 的符号发生改变。综合练习会考查你对这一细微区别的理解。
8. Optimisation Problems | 优化问题
These real‑ pression for a quantity such as area or volume, then differentiating to find its maximum or minimum. Always express the quantity in terms of a single variable, differentiate, set f' ationary point. Remember to check that your answer makes sense within the given constraints.
这类实际问题需要先列出面积或体积等量的表达式，然后通过求导找出最大值或最小值。始终把该量表示成单一变量的函数，求导，令 f'(x) = 0，并验证驻点性质。记得检查答案是否在给定约束条件下合理。
9. Sketching the Gradient Function | 绘制导函数的图像
You may be given the graph of f(x) and asked to sketch f'(x). Key features: where f(x) has a turning point, f'(x) crosses the x‑axis; where f(x) is increasing, f'(x) is positive; where f(x) is decreasing, f'(x) is negative. The steepness of f(x) influences the magnitude of f'(x). Practice linking the two graphs visually.
题目可能会给出 f(x) 的图像并要求绘制 f'(x) 的草图。关键特征：f(x) 有极值点处，f'(x) 穿过 x 轴；f(x) 递增时 f'(x) 为正；f(x) 递减时 f'(x) 为负。f(x) 的陡峭程度影响 f'(x) 的大小。练习将两者图像在视觉上联系起来。
10. Linking Position, Velocity and Acceleration | 位移、速度与加速度的联系
ime when a particle is at rest (v = 0), or the distance travelled. Differentiate carefully and interpret the physical meaning of any stationary points you find.
当给出位移函数 s(t) 时，速度 v(t) = ds/dt，加速度 a(t) = dv/dt = d²s/dt²。题目可能需要求最大速度、质点静止的时刻（v = 0）或运动距离。仔细求导并解释所找到驻点的物理意义。
11. Mixed Problem Solving – Combining Concepts | 混合解题 —— 综合多个概念
The most challenging questions in Mixed Exercise 12 require you to combine several techniques. For example, you might first need to find a tangent equation, then locate where it meets the curve again, or use a stationary point condition to find an unknown constant, and then determine the nature of other turning points. Read each problem carefully and identify which differentiation tools are needed.
综合练习 12 中最有挑战性的题目需要你结合多种技巧。例如，你可能先要求出一条切线方程，再找到它与曲线的另一个交点；或者利用驻点条件求出未知常数，然后判断其他极值点的性质。仔细读题，确定需要用到哪些微分工具。
12. Common Pitfalls and Exam Tips | 常见陷阱与考试提示
Watch out for forgetting to differentiate constant terms, mishandling negative or fractional powers, and confusing tangents with normals. Always show the derivative before substituting coordinates. If you are asked to prove a point is a maximum, give clear evidence using the second derivative or a gradient table. Check your final answers for arithmetic errors and ensure they are simplified fully.
注意不要忘记常数项求导为零、错误处理负指数或分数指数、混淆切线与法线。先写出导函数再代入坐标。如果要求证明某点是最大值，要用二阶导数或梯度表给出明确证据。检查最终答案是否有算术错误，并确保已完全化简。
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