Exercise 6A: Introduction to Differentiation | 练习6A：微分入门

📚 Exercise 6A: Introduction to Differentiation | 练习6A：微分入门

Exercise 6A in the IB Mathematics course builds the essential skills for understanding how functions change. This exercise introduces the concept of the derivative from first principles, the power rule, and basic techniques for differentiating polynomial functions. Mastering these problems will lay a solid foundation for studying tangents, normals, rates of change, and optimisation later in the syllabus.

IB数学课程中的练习6A旨在培养理解函数变化方式的基本技能。本练习介绍了从第一原理出发的导数概念、幂法则以及多项式函数求导的基本技巧。掌握这些题目将为后续学习切线、法线、变化率和最优化等内容奠定坚实基础。

1. Understanding the Derivative from First Principles | 从第一原理理解导数

The derivative of a function f(x) at any point x measures the instantaneous rate of change, which corresponds to the gradient of the tangent line. The formal definition uses a limit of the difference quotient as h approaches zero.

函数 f(x) 在任意点 x 处的导数衡量的是瞬时变化率，即切线的斜率。正式定义使用了当 h 趋近于零时差商的极限。

f'(x) = lim_h→0 [f(x + h) − f(x)] / h

f'(x) = lim_h→0 [f(x + h) − f(x)] / h

anding (x + h)², simplifying the numerator, and then cancelling h before taking the limit. The same process works for any function, but the algebra can become heavy, which is why shortcut rules are later introduced.

将这一定义应用于 f(x) = x² 这样的简单二次函数，需要展开 (x + h)²，化简分子，然后在取极限之前消去 h。同样的过程适用于任何函数，但代数运算可能变得繁重，这也是后来引入快捷求导法则的原因。

---

2. The Power Rule for Differentiation | 微分的幂法则

The power rule states that if f(x) = xⁿ, where n is any real number, then f' ⁻¹. This rule is derived from first principles and dramatically speeds up differentiation.

幂法则指出，若 f(x) = xⁿ（其中 n 为任意实数），则 f'(x) = n xⁿ⁻¹。该法则由第一原理推导而来，并能极大加快求导速度。

The most common applications in Exercise 6A involve integer exponents, such as x³, x⁵, or x⁻². Make sure to subtract one from the exponent and multiply by the original exponent, even when the exponent is negative or a fraction.

练习6A中最常见的应用涉及整数指数，例如 x³、x⁵ 或 x⁻²。无论指数为负数还是分数，都要确保将指数减一并乘以原来的指数。

Function f(x)	Derivative f'(x)
x²	2x
x⁵	5x⁴
x⁻³	−3x⁻⁴
x½	(½)x−½

---

3. Differentiating Sums and Differences | 求导和与差

Differentiation is a linear operation, meaning that the derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives. You can work term by term with polynomials.

求导是一种线性运算，即和的导数为导数之和，差的导数为导数之差。对于多项式，你可以逐项进行求导。

you differentiate each power of x separately: the derivative is 12x³ − 15x² + 2. The constant −7 differentiates to zero because it has no x-dependence.

例如，若 f(x) = 3x⁴ − 5x³ + 2x − 7，你可以对每一项 x 的幂分别求导：导数为 12x³ − 15x² + 2。常数项 −7 求导为零，因为它不依赖于 x。

Always remember to keep the coefficients exactly as they appear before applying the power rule. Common errors arise when students forget to multiply the coefficient by the exponent.

切记在应用幂法则之前保留系数原样。常见错误是学生忘记将系数与指数相乘。

---

4. Derivatives of Constant Functions | 常数函数的导数

s a real number, has a derivative of zero for all x. Graphically, the line is horizontal, so its gradient is zero everywhere.

常数函数 f(x) = c（c 为实数）对一切 x 的导数均为零。从图形来看，该直线是水平的，因此其斜率处处为零。

n: f(x + h) − f(x) = c − c = 0, so the difference quotient is 0/h = 0, and the limit is 0.

这一结果可从第一原理定义来理解：f(x + h) − f(x) = c − c = 0，因此差商为 0/h = 0，极限为 0。

In many Exercise 6A problems, constants appear as standalone terms or as coefficients in front of other functions. Knowing that the derivative of a constant is zero simplifies calculations significantly.

在练习6A的许多问题中，常数可以作为独立项出现，也可以作为其他函数前面的系数。了解常数的导数为零可极大地简化计算。

---

5. Finding the Gradient at a Specific Point | 求特定点的梯度

Once you have obtained f'(x), the derivative function, you can evaluate it at a particular x-value to find the gradient of the tangent at that point. This is a key skill tested in these exercises.

一旦得到导数函数 f'(x)，你就可以代入特定的 x 值，求出该点处切线的斜率。这是这些练习中考查的一项关键技能。

If f(x) = x³ − 2x, then f'(x) = 3x² − 2. To find the gradient at x = 1, substitute x = 1 into f'(x): f' 3(1)² − 2 = 1. This tells you the curve is sloping upwards with gradient 1 at that point.

若 f(x) = x³ − 2x，则 f'(x) = 3x² − 2。要求 x = 1 处的梯度，可将 x = 1 代入 f'(x)：f'(1) = 3(1)² − 2 = 1。这表明曲线在该点处以斜率 1 向上倾斜。

Mark schemes frequently award marks for correctly finding the derivative and then substituting the coordinate. Always show these two steps clearly.

评分方案通常会对正确求出导数并随后代入坐标的步骤给分。务必清晰地展示这两个步骤。

---

广告 / Sponsored

6. The Concept of Tangent and Normal Lines | 切线与法线的概念

The tangent to a curve at a point has a gradient equal to the derivative at that point. The normal line is perpendicular to the tangent, so its gradient is the negative reciprocal of the tangent's gradient.

曲线在某点处的切线，其斜率等于该点处的导数值。法线垂直于切线，因此其斜率为切线斜率的负倒数。

If the gradient of the tangent is m_t, then the gradient of the normal m_n satisfies m_t × m_n = −1, provided m_t ≠ 0.

如果切线的斜率为 m_t，则法线的斜率 m_n 满足 m_t × m_n = −1，前提是 m_t ≠ 0。

Even though Exercise 6A focuses mainly on computing derivatives, a few questions may ask you to state the gradient of the tangent at a point, linking directly to this geometric interpretation.

尽管练习6A主要侧重于计算导数，但有少数问题可能会要求你写出某点处切线的斜率，这直接与这种几何解释相关联。

---

7. Applying the Derivative to Simple Polynomials | 将导数应用于简单多项式

Most problems in Exercise 6A involve differentiating polynomials of degree up to 5 or 6. You must be comfortable handling fractional and negative exponents, as they appear when rewriting expressions like 1/x² or √x.

练习6A中的大多数问题涉及对高达5次或6次的多项式进行求导。你必须熟练处理分数指数和负指数，因为它们在重写像 1/x² 或 √x 这样的表达式时会出现。

Rewrite 1/x² as x⁻², so its derivative is −2x⁻³. Similarly, √x can be written as x^½, and its derivative is (½)x^−½. This algebraic preparation is essential before applying the power rule.

将 1/x² 重写为 x⁻²，其导数为 −2x⁻³。同样，√x 可写作 x^½，其导数为 (½)x^−½。在应用幂法则之前，这种代数准备是不可或缺的。

ant and a function is differentiated by keeping the constant factor: if f(x) = 4x³, then f' 4 × 3x² = 12x². Never differentiate the constant factor as if it were a separate variable.

常数与函数的乘积求导时保留常数因子：若 f(x) = 4x³，则 f'(x) = 4 × 3x² = 12x²。切勿将常数因子当作独立变量来求导。

---

8. Higher-Order Derivatives | 高阶导数

The second derivative, denoted f''(x) or d²y/dx², is the derivative of the first derivative. It describes the rate of change of the gradient and is used to determine the concavity of a function.

二阶导数记作 f''(x) 或 d²y/dx²，是一阶导数的导数。它描述了梯度的变化率，用于判断函数的凹凸性。

For a polynomial, you can simply differentiate once to get f'(x) and then differentiate again using the same rules. For example, if f(x) = 2x⁴, then f'(x) = 8x³ and f''(x) = 24x².

对于多项式，你只需先求一次导得到 f'(x)，然后再用同样的法则求一次导。例如，若 f(x) = 2x⁴，则 f'(x) = 8x³，而 f''(x) = 24x²。

In Exercise 6A, higher-order derivative questions are straightforward and meant to reinforce the idea that differentiation can be repeated. Always simplify each derivative before moving to the next one.

在练习6A中，高阶导数问题都很直接，旨在强化求导可以反复进行这一概念。每次求导后都要先化简，再进行下一次求导。

---

9. Common Mistakes to Avoid | 常见错误避免

One frequent error is forgetting to multiply the coefficient by the exponent. For 5x⁴, the derivative is 20x³, not 5x³. Another mistake is mishandling the exponent when it becomes negative; practice until subtracting one becomes automatic.

一个常见错误是忘记将系数乘以指数。对于 5x⁴，导数是 20x³，而不是 5x³。另一个错误是在指数变为负数时处理不当；要通过练习使指数减一变得自动化。

Students also sometimes confuse the derivative of a constant with the constant itself. The derivative of 7 is 0, not 7. Also, do not add extra terms when differentiating sums; differentiate each term individually.

学生有时还会混淆常数的导数与常数本身。7 的导数是 0，而不是 7。此外，在对和求导时不要添加多余项；应逐项单独求导。

Writing notation clearly is crucial: use f'(x) or dy/dx consistently and never omit the variable when specifying f'(x) at a point. Sloppy notation can lose marks in examinations even if the numerical answer is correct.

清楚地书写符号至关重要：始终一致地使用 f'(x) 或 dy/dx，并且在指定某点的 f'(x) 时切勿省略变量。即使数值答案正确，潦草的符号也可能在考试中丢分。

---

10. Worked Examples from Exercise 6A | 练习6A的例题解答

em: Differentiate f(x) = 6x⁵ − 3x² + 8x − 10. Applying the power rule to each term gives f'(x) = 30x⁴ − 6x + 8. The constant term −10 disappears.

让我们看一个典型问题：求 f(x) = 6x⁵ − 3x² + 8x − 10 的导数。对每一项应用幂法则，得到 f'(x) = 30x⁴ − 6x + 8。常数项 −10 消失了。

. Rewrite as y = 2x⁻¹, then dy/dx = −2x⁻² = −2/x². At x = 4, the gradient is −2/16 = −1/8.

另一个例子：求曲线 y = 2/x 在 x = 4 处的梯度。重写为 y = 2x⁻¹，则 dy/dx = −2x⁻² = −2/x²。在 x = 4 处，梯度为 −2/16 = −1/8。

If asked for the second derivative of f(x) = x³ + 3x, find f'(x) = 3x² + 3, then f'' lways check your powers and coefficients after each step to avoid carrying errors forward.

如果要求 f(x) = x³ + 3x 的二阶导数，先求 f'(x) = 3x² + 3，然后 f''(x) = 6x。每步之后都要检查指数和系数，以避免错误前移。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）