The Derivative of xⁿ for Positive Integer Powers | 正整数幂函数xⁿ的导数

📚 The Derivative of xⁿ for Positive Integer Powers | 正整数幂函数xⁿ的导数

In IB Mathematics, the derivative of xⁿ is one of the most fundamental results in calculus. For any positive integer n, the derivative is n xⁿ⁻¹. This article explains why this rule works, derives it from first principles, and applies it to exam-style problems.

在IB数学中，xⁿ的导数是最基础的结果之一。对于任何正整数n，其导数都是 n xⁿ⁻¹。本文将解释这条规则为何成立，从第一性原理出发进行推导，并应用于考试风格的问题。

1. The Definition of the Derivative | 导数的定义

Before we can differentiate xⁿ, we must recall the formal definition of the derivative. For a function f, its derivative at x is the limit of the average rate of change over an interval that shrinks to zero.

在微分xⁿ之前，我们必须回顾导数的正式定义。对于函数f，它在x处的导数是区间缩小至零时平均变化率的极限。

f′(x) = lim (h→0) [f(x+h) − f(x)] / h

This limit is often called differentiating from first principles. If the limit exists, f is said to be differentiable at x.

这个极限通常称为”从第一性原理求导”。如果极限存在，就称f在x处可导。

2. Expanding (x + h)ⁿ | 展开 (x + h)ⁿ

The key step in differentiating xⁿ from first principles is to expand (x + h)ⁿ. We use the binomial theorem, which expresses the power as a sum of terms with decreasing powers of x.

从第一性原理微分xⁿ的关键步骤是展开(x + h)ⁿ。我们使用二项式定理，将该幂表示为x的降幂项之和。

(x + h)ⁿ = xⁿ + n xⁿ⁻¹ h + [n(n − 1) / 2] xⁿ⁻² h² + … + hⁿ

For example, when n = 3, the expansion is (x + h)³ = x³ + 3x²h + 3xh² + h³.

例如，当n = 3时，展开式为(x + h)³ = x³ + 3x²h + 3xh² + h³。

3. Deriving the General Formula | 推导一般公式

Substitute the expansion into the first-principles definition. The first term xⁿ cancels with −f(x), leaving n xⁿ⁻¹ h plus terms that each contain at least h².

将展开式代入第一性原理的定义。第一项xⁿ与−f(x)相消，留下 n xⁿ⁻¹ h 加上每一项至少含有h²的后续项。

After dividing by h, we obtain n xⁿ⁻¹ plus terms involving h. As h approaches zero, these terms vanish.

除以h后，我们得到 n xⁿ⁻¹ 加上含有h的项。当h趋近于零时，这些项消失。

d/dx (xⁿ) = n xⁿ⁻¹

This result is the power rule for positive integer exponents.

这个结果就是正整数指数的幂法则。

4. Applying the Power Rule | 应用幂法则

Now that we have the formula, we can differentiate any positive integer power quickly. The table below shows some common cases.

现在我们有了公式，就能快速微分任何正整数幂。下表展示了一些常见情形。

f(x)	f′(x)
x¹	1
x²	2x
x³	3x²
x⁵	5x⁴
x¹⁰	10x⁹

Notice that x⁰ = 1 for x ≠ 0, so the formula works even when n = 1.

注意x⁰ = 1（x ≠ 0），所以公式在n = 1时同样成立。

5. Combining with Constant Multiples | 常数倍与幂法则结合

Often the function you need to differentiate is multiplied by a constant. The constant multiple rule states that the derivative of c·f(x) is c·f′(x).

通常需要微分的函数会乘以一个常数。常数倍法则指出，c·f(x)的导数是c·f′(x)。

d/dx (c xⁿ) = c n xⁿ⁻¹

If y = 4x², then dy/dx = 4 × 2x = 8x.

若y = 4x²，则dy/dx = 4 × 2x = 8x。
If y = −3x⁶, then dy/dx = −18x⁵.

若y = −3x⁶，则dy/dx = −18x⁵。

6. The Sum and Difference Rules | 和差法则

When a function is a sum or difference of power terms, differentiate term by term.

当函数是幂项的和或差时，逐项求导。

d/dx (u + v) = du/dx + dv/dx, d/dx (u − v) = du/dx − dv/dx

For example, f(x) = x⁴ + x² has f′(x) = 4x³ + 2x.

例如，f(x) = x⁴ + x² 的导数为 f′(x) = 4x³ + 2x。

This rule works because the derivative is a linear operator; the limit of a sum is the sum of the limits.

该规则之所以成立，是因为导数是一种线性算子；和的极限等于极限的和。

7. Tangents and Normals | 切线与法线

One of the main uses of the derivative is finding the equation of the tangent line to a curve at a given point.

导数的主要用途之一是在给定点处求曲线的切线方程。

If y = xⁿ, the slope of the tangent at x = a is n aⁿ⁻¹. The tangent line then has equation y − aⁿ = n aⁿ⁻¹ (x − a).

若y = xⁿ，在x = a处切线的斜率为 n aⁿ⁻¹。因此切线方程为 y − aⁿ = n aⁿ⁻¹ (x − a)。

The normal line is perpendicular to the tangent, so its slope is the negative reciprocal of the tangent’s slope.

法线与切线垂直，因此其斜率为切线斜率的负倒数。

8. Higher-Order Derivatives | 高阶导数

Applying the power rule repeatedly gives higher-order derivatives.

重复应用幂法则可以得到高阶导数。

f(x) = x⁷ gives f′(x) = 7x⁶.

f(x) = x⁷ 得到 f′(x) = 7x⁶。
Then f″(x) = 42x⁵.

再求导得 f″(x) = 42x⁵。
And f‴(x) = 210x⁴.

三阶导数 f‴(x) = 210x⁴。

In general, the k-th derivative of xⁿ is n(n−1)···(n−k+1) xⁿ⁻ᵏ.

一般地，xⁿ的k阶导数为 n(n−1)···(n−k+1) xⁿ⁻ᵏ。

9. Common Pitfalls | 常见误区

Forgetting to subtract one from the exponent. The new exponent is n − 1, not n.

忘记指数减一。新指数是n − 1，而不是n。
Forgetting to multiply by the original exponent. Always include the factor n.

忘记乘以原指数。必须包含因子n。
Confusing xⁿ with exponential functions. The power rule applies to constant exponents, not to aⁿ.

将xⁿ与指数函数混淆。幂法则适用于常数指数，而不是aⁿ。
Using the product rule unnecessarily. For c xⁿ, use the constant multiple rule first.

不必要地使用乘积法则。对于c xⁿ，应先使用常数倍法则。

10. Exam-Style Practice | 考试风格练习

Let us solve a typical IB question. Find the derivative of f(x) = 5x³ − 2x⁴ + x² − 7.

让我们解决一道典型的IB题目。求 f(x) = 5x³ − 2x⁴ + x² − 7 的导数。

Apply the sum and difference rules term by term:

逐项应用和差法则：

f′(x) = 15x² − 8x³ + 2x

The constant −7 has derivative zero. Always double-check by re-reading the question and confirming the exponents.

常数−7的导数为零。务必重读题目并确认指数来检查答案。

Another common question asks for the gradient of y = x⁸ at x = 1. The gradient is 8·1⁷ = 8.

另一个常见问题是求 y = x⁸ 在 x = 1 处的梯度。梯度为 8·1⁷ = 8。

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The Derivative of xⁿ for Positive Integer Powers | 正整数幂函数xⁿ的导数