A-Level Mathematics: Term-by-Term Differentiation of Polynomial Functions | A-Level数学：多项式函数的逐项求导

📚 A-Level Mathematics: Term-by-Term Differentiation of Polynomial Functions | A-Level数学：多项式函数的逐项求导

Differentiation is a core operation in A-Level Mathematics. For polynomial functions, the process is wonderfully simple: you differentiate each term separately and add the results. This method, known as term-by-term differentiation, is powerful, efficient, and forms the foundation for almost every calculus problem you will meet.

微分是A-Level数学中的核心运算。对于多项式函数，微分过程非常简单：分别对每一项求导，再将结果相加。这种方法被称为“逐项求导”，它强大、高效，并且是几乎所有后续微积分问题的基础。

---

1. The Power Rule | 幂法则

The power rule states that if f(x) = xⁿ , where n is any real constant, then the derivative is f'(x) = nx^n-1 . In words, bring the power down in front, then reduce the power by one.

幂法则指出：若 f(x) = xⁿ ，其中 n 是任意实数常数，则导数为 f'(x) = nx^n-1 。换言之，将指数移到前面，然后将指数减一。

d/dx (xⁿ) = nxⁿ⁻¹

For example, the derivative of x⁵ is 5x⁴ , and the derivative of x^1/2 is 12x^-1/2 . The rule works for positive, negative, and fractional powers alike.

例如， x⁵ 的导数是 5x⁴ ， x^1/2 的导数是 12x^-1/2 。该法则对正指数、负指数和分数指数同样适用。

---

2. The Constant Rule | 常数法则

A constant term, such as 7, -3, or π, has a derivative of zero. This makes sense geometrically: the graph of a constant is a horizontal line, whose slope is always zero.

常数项，如7、-3或π，其导数为零。这在几何上显而易见：常数的图像是一条水平直线，其斜率始终为零。

d/dx (c) = 0, where c is a constant

When differentiating a polynomial, do not forget to differentiate every constant term. Even if the constant is written as 52 or √2, its derivative is still 0.

对多项式求导时，不要忘记处理每一个常数项。即使常数写成 52 或 √2，其导数仍然为0。

---

3. The Constant Multiple Rule | 常数倍法则

If a term has a constant coefficient, you simply keep the coefficient and differentiate the variable part. For example, the derivative of 4x³ is 4 × 3x² = 12x² .

如果某一项带有常数系数，只需保留该系数并对其变量部分求导。例如， 4x³ 的导数为 4 × 3x² = 12x² 。

d/dx (c·xⁿ) = c·n·xⁿ⁻¹

This rule is often combined with the power rule without comment. It is essential for handling polynomials before applying the sum rule.

这个法则通常与幂法则结合使用，不再单独说明。它是在使用和差法则之前处理多项式各项的关键步骤。

---

4. The Sum and Difference Rules | 和差法则

The sum rule states that the derivative of a sum of functions equals the sum of their derivatives. Similarly, the derivative of a difference equals the difference of their derivatives.

和法则指出：函数之和的导数等于各函数导数之和。类似地，函数之差的导数等于各函数导数之差。

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

Because polynomials are sums of power functions, these two rules together justify term-by-term differentiation. You never need to differentiate a whole polynomial at once; just attend to each term individually.

由于多项式是幂函数的和，这两个法则共同为逐项求导提供了依据。你永远不需要一次性对整个多项式求导，只需逐项处理即可。

---

5. Term-by-Term Differentiation in Practice | 逐项求导的实际操作

To differentiate a polynomial, rewrite it in standard form a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, then differentiate each term using the power rule and constant multiple rule. Finally, add the derived terms together.

对多项式求导时，先将其写成标准形式 a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀，然后利用幂法则和常数倍法则对每一项分别求导。最后将各个导数相加即可。

Example: Find the derivative of f(x) = 3x⁴ - 5x² + 7x - 11 .

例：求 f(x) = 3x⁴ - 5x² + 7x - 11 的导数。

f'(x) = 12x³ - 10x + 7

Notice that the constant -11 disappears in the derivative, and the derivative of 7x is simply 7, since 7x = 7x¹.

注意常数 -11 在导数中消失，而 7x 的导数就是7，因为 7x = 7x¹。

---

6. Simplifying Terms Before Differentiation | 求导前的化简

Sometimes a polynomial is not given in standard powers. Terms like ¹⁄_x² , √x , or ^{x3 + 2x}⁄_x should be rewritten using negative and fractional exponents first.

有时多项式并非以标准幂形式给出。像 ¹⁄_x² 、 √x 或 ^{x3 + 2x}⁄_x 这样的项，应先改写为负指数或分数指数形式。

• ¹⁄_x² = x^-2

  ¹⁄_x² = x^-2

• √x = x^1/2

  √x = x^1/2

• ^{x3 + 2x}⁄_x = x² + 2

  ^{x3 + 2x}⁄_x = x² + 2

After rewriting, apply the power rule. For example, ¹⁄_x² becomes -2x^-3 = -²⁄_x³ .

改写后应用幂法则。例如， ¹⁄_x² 变为 -2x^-3 = -²⁄_x³ 。

---

7. Differentiating Polynomials with Negative and Fractional Powers | 含负指数与分数指数的多项式求导

Consider g(x) = 2x³ + 3x^-1 + x^1/2 . Differentiating term-by-term gives g'(x) = 6x² - 3x^-2 + 12x^-1/2 .

考虑 g(x) = 2x³ + 3x^-1 + x^1/2 。逐项求导得到 g'(x) = 6x² - 3x^-2 + 12x^-1/2 。

Be careful with signs: 3x^-1 has derivative 3 × (-1)x^-2 = -3x^-2. Negative powers keep the power rule exactly the same, but errors in sign are common.

注意符号：3x^-1 的导数为 3 × (-1)x^-2 = -3x^-2。负指数完全遵循幂法则，但符号错误很常见。

---

广告 / Sponsored

8. Second and Higher Derivatives | 二阶及高阶导数

The second derivative f''(x) is obtained by differentiating f'(x) again. For a polynomial of degree n, each differentiation reduces the degree by 1, so after n+1 differentiations the result is zero.

二阶导数 f''(x) 是对 f'(x) 再次求导得到的。对于 n 次多项式，每求导一次次数减1，因此经过 n+1 次求导后结果为零。

Example: If f(x) = x⁴ - 2x³ , then f'(x) = 4x³ - 6x² , f''(x) = 12x² - 12x , f'''(x) = 24x - 12 , and f''''(x) = 24 .

例：若 f(x) = x⁴ - 2x³ ，则 f'(x) = 4x³ - 6x² ， f''(x) = 12x² - 12x ， f'''(x) = 24x - 12 ，且 f''''(x) = 24 。

Higher derivatives are used in curve sketching, kinematics, and Taylor series approximations. In A-Level, the second derivative is especially important for classifying stationary points.

高阶导数用于曲线描绘、运动学以及泰勒级数近似。在A-Level中，二阶导数对于判断驻点性质尤为重要。

---

9. Applications: Tangents and Normals | 应用：切线与法线

Given a polynomial y = f(x), the derivative f'(a) gives the gradient of the tangent line at x = a. The normal line is perpendicular to the tangent, so its gradient is -1f'(a), provided f'(a) ≠ 0.

给定多项式 y = f(x)，导数 f'(a) 给出在 x = a 处切线的斜率。法线垂直于切线，因此其斜率为 -1f'(a)，前提是 f'(a) ≠ 0。

Example: Find the equation of the tangent to y = x² + 3x at x = 1.

例：求曲线 y = x² + 3x 在 x = 1 处的切线方程。

y = 5x - 1

The derivative is 2x + 3, so the gradient at x = 1 is 5. The point is (1, 4), giving the tangent equation y - 4 = 5(x - 1).

导数为 2x + 3，因此在 x = 1 处的斜率为5。该点为(1, 4)，切线方程为 y - 4 = 5(x - 1)。

---

10. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Term-by-term differentiation is simple, but small slips cause many lost marks. Here are the most frequent pitfalls:

逐项求导虽然简单，但小失误常导致丢分。以下是最常见的陷阱：

Mistake	Correct	Explanation
3x4 → 12x3	3x4 → 12x3	Correct multiplication: 3 × 4 = 12.
x-2 → -2x-1	x-2 → -2x-3	Subtract 1 from the exponent: -2 - 1 = -3.
5 → 5	5 → 0	Constants always differentiate to zero.

Always write out each term's derivative before adding them together. This habit reduces sign errors and makes your working clear to examiners.

务必先写出每一项的导数，再合并相加。这个习惯能减少符号错误，并让阅卷老师看清你的思路。

---

11. Worked Examination-Style Problems | 考试风格例题

Problem 1: Differentiate f(x) = 5x³ - 2x² + 4x - 7 .

题目1：求 f(x) = 5x³ - 2x² + 4x - 7 的导数。

f'(x) = 15x² - 4x - 4x⁻²

Rewrite 4x as 4x^-1, then differentiating gives -4x^-2. The constant -7 vanishes.

将 4x 改写为 4x^-1，求导得 -4x^-2。常数 -7 消失。

Problem 2: Find the gradient of y = 2x² - 3x + 1 at the point where x = 2.

题目2：求 y = 2x² - 3x + 1 在 x = 2 处的斜率。

Gradient = 5

Since y' = 4x - 3, substituting x = 2 gives 4(2) - 3 = 5.

因为 y' = 4x - 3，代入 x = 2 得 4(2) - 3 = 5。

---

12. Summary | 总结

Term-by-term differentiation turns polynomial calculus into a sequence of simple steps: apply the power rule to each term, keep coefficients, ignore constants, and combine results. Mastery of this technique unlocks the rest of differential calculus.

逐项求导将多项式微积分转化为一系列简单步骤：对每一项应用幂法则、保留系数、忽略常数项、合并结果。掌握这一技巧将为后续的微分学内容打开大门。

Remember the two core operations: multiply by the power, then reduce the power by one. Practice until the process becomes automatic, and always check that you have accounted for every term in the original polynomial.

牢记两个核心操作：乘以指数，然后将指数减一。反复练习直到这一过程驾轻就熟，并始终检查原多项式中的每一项都被正确处理。

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）