Exercise 15A: Differentiation Practice | 练习15A：微分练习

📚 Exercise 15A: Differentiation Practice | 练习15A：微分练习

Welcome to Exercise 15A, a carefully structured set of problems that sharpens your differentiation skills for the IB Mathematics course. In this article, we break down the core rules, techniques, and common pitfalls you will encounter while working through these exercises. Whether you are studying Analysis and Approaches or simply reinforcing your calculus toolkit, mastering these drills will build the fluency required for higher-level applications like optimisation, kinematics, and implicit differentiation.

欢迎来到练习15A，这是一套经过精心设计的题目，旨在磨炼你在IB数学课程中的微分技巧。在这篇文章中，我们将解析你在完成这些练习时会遇见的各种核心法则、技巧与常见误区。无论你正在学习分析与方法课程，还是仅仅在强化微积分工具，掌握这些练习将为优化、运动学以及隐函数求导等高阶应用打下必要的基础。

1. Understanding the Derivative | 理解导数

point gives the slope of the tangent line, which represents the instantaneous rate of change. In Exercise 15A, you will be asked to find derivatives from first principles using the limit definition: f' hough later exercises favour rules, revisiting the definition deepens your conceptual understanding.

函数 f(x) 在某一点的导数给出切线的斜率，代表着瞬间变化率。在练习15A中，你会被要求利用极限定义 f'(x) = lim(h→0) [f(x+h) − f(x)] / h 从第一原理求导。虽然后续练习倾向于使用求导法则，但重温定义能加深你对概念的理解。

2. Power Rule | 幂法则

The most frequently used tool in Exercise 15A is the Power Rule: if f(x) = xⁿ, then f' t also to negative and fractional exponents. You must rewrite radical expressions, such as √x, as x½ before differentiating.

练习15A中最常用的工具就是幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。该法则不仅适用于正整数次幂，也同样适用于负指数和分数指数。在求导前务必将根式表达式（如 √x）改写为 x½。

d/dx (x⁴) = 4x³
d/dx ( 1/x² ) = d/dx (x⁻²) = −2x⁻³
d/dx (√x) = ½ x⁻½

3. Sum and Difference Rule | 和差法则

simply apply the power rule to each monomial. The rule states that d/dx [f(x) ± g(x)] = f'(x) ± g'(x).

微分是线性的，这意味着你可以逐项求导。在练习15A中，你会遇到多项式表达式，只需对每一个单项式应用幂法则。法则指出 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。

4. Product Rule | 乘积法则

ble functions, say u(x) · v(x), the product rule must be used. The rule is: d/dx (u·v) = u'·v + u·v'. In Exercise 15A, you may see products like x² sin x or (3x+1) eˣ, where choosing the order of u and v carefully helps simplify the result.

当一个函数表示为两个可微函数的乘积（例如 u(x)·v(x)）时，就必须使用乘积法则。该法则为：d/dx (u·v) = u'·v + u·v'。在练习15A中，你可能会看到诸如 x² sin x 或 (3x+1)eˣ 的产品，仔细选择 u 和 v 会有助于简化结果。

5. Quotient Rule | 商法则

For a function written as a quotient u/v, the derivative is given by (u'·v − u·v') / v². Many students struggle with the minus sign and the correct order. Exercise 15A contains rational functions that test your ability to apply this rule accurately, such as (x²+1)/(x-2). Alternatively, you can sometimes rewrite the expression using negative exponents and use the product rule.

对于写成商形式 u/v 的函数，其导数为 (u'·v − u·v') / v²。许多学生容易在减号和先后顺序上出错。练习15A中包含有理函数以检验你精确应用此法则的能力，例如 (x²+1)/(x-2)。有时你也可以将表达式用负指数改写，再使用乘积法则。

d/dx ( (x²+1)/(x-2) ) = [2x(x-2) − (x²+1)·1] / (x-2)²

6. Chain Rule | 链式法则

: dy/dx = dy/du · du/dx. In Exercise 15A, you will differentiate functions like sin(3x), e^2x², and ln(5x+1). Identifying the inner function u is the key step. Mastering the chain rule early will pay off in later topics such as related rates and implicit differentiation.

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链式法则对于复合函数求导不可或缺：dy/dx = dy/du · du/dx。在练习15A中，你将求导如 sin(3x)、e^2x² 和 ln(5x+1) 等函数。识别内层函数 u 是关键步骤。尽早掌握链式法则，会在后期相关变化率与隐函数求导等课题中让你受益匪浅。

7. Derivatives of Trigonometric Functions | 三角函数的导数

than x, the chain rule must be applied immediately. For example, d/dx (sin 2x) = 2 cos 2x.

练习15A包含必须熟记的基本三角函数导数：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。当这些函数的自变量不是 x 时，必须立刻运用链式法则。例如，d/dx (sin 2x) = 2 cos 2x。

8. Derivatives of Exponential and Logarithmic Functions | 指数与对数函数的导数

The natural exponential function eˣ has the remarkable property d/dx (eˣ) = eˣ. For natural logarithms, d/dx (ln x) = 1/x. When the base is not e, also use d/dx (aˣ) = aˣ a. Exercise 15A often combines these with the chain rule, leading to expressions like 2e³ˣ or 1/(x ln 10).

自然指数函数 eˣ 拥有一个非凡的性质：d/dx (eˣ) = eˣ。对于自然对数，d/dx (ln x) = 1/x。当底数不为 e 时，还需用到 d/dx (aˣ) = aˣ ln a。练习15A经常将它们与链式法则结合，从而得到 2e³ˣ 或 1/(x ln 10) 等表达式。

9. Implicit Differentiation | 隐函数求导

curves like x² + y² = 25, where you obtain dy/dx = −x/y through implicit differentiation.

当一个方程将 y 隐式地定义为 x 的函数而无法显式解出 y 时，可以对方程两边同时对 x 求导，将 y 视为隐函数并运用链式法则。练习15A可能会引入圆或诸如 x² + y² = 25 的曲线，此时通过隐函数求导可得 dy/dx = −x/y。

10. Higher-Order Derivatives | 高阶导数

The second derivative f''(x) is simply the derivative of the first derivative. It measures the rate of change of the slope and indicates concavity. Exercise 15A asks you to compute second and sometimes third derivatives for polynomials, exponentials, and trigonometric functions. Correct notation is vital: d²y/dx² or f''(x).

二阶导数 f''(x) 就是一阶导数的导数。它衡量斜率的变化率并反映凹凸性。练习15A要求你计算多项式、指数函数和三角函数等的二阶导数，有时甚至是三阶导数。正确使用记号至关重要：d²y/dx² 或 f''(x)。

11. Tangents and Normals | 切线与法线

Once you have the derivative f' oint x = a, finding the equation of the tangent line becomes straightforward: y − f(a) = f'(a)(x − a). The normal line is perpendicular, with slope −1/f' ten requiring you to evaluate the function and its derivative at a specified x-coordinate.

一旦得到点 x = a 处的导数 f'(a)，求切线方程就变得简单了：y − f(a) = f'(a)(x − a)。法线与切线垂直，其斜率为 −1/f'(a)。练习15A中含有要求同时求出切线和法线方程的问题，通常需要计算函数及其导数在给定 x 坐标处的值。

12. Common Mistakes in Exercise 15A | 练习15A常见错误

Even a strong student can slip up by forgetting to apply the chain rule, mishandling minus signs in the quotient rule, or confusing the derivative of cos x with that of sin x. Another typical error is misplacing parentheses when substituting into the product rule. Always double-check your work and practise with the varied examples in Exercise 15A to make these rules second nature.

即使基础扎实的学生也可能因忘记应用链式法则、在商法则中处理错负号或混淆 cos x 与 sin x 的导数而失误。另一个典型错误是在代入乘积法则时括号位置出错。请务必仔细检查，并通过练习15A中多样的例题反复训练，让这些法则成为你的第二天性。

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