Differentiation of Power Functions xⁿ: Formula and Applications | 幂函数 xⁿ 的求导公式及应用

📚 Differentiation of Power Functions xⁿ: Formula and Applications | 幂函数 xⁿ 的求导公式及应用

Power functions are among the most fundamental building blocks in calculus. The rule for differentiating xⁿ appears at the very beginning of any derivative course and remains essential for solving problems in geometry, physics, economics and beyond. In this article we will derive the rule, examine special cases, and explore its most common applications in A-level and IB-style mathematics.

幂函数是微积分中最基础的部分之一。对 xⁿ 求导的法则出现在导数课程的最开头，并且在几何、物理、经济等许多领域的解题中始终至关重要。本文将推导该法则，讨论特殊情形，并探究其在 A-level、IB 等考试中最常见的应用。

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1. What Is a Power Function? | 什么是幂函数？

A power function has the form f(x) = xⁿ, where x is a variable and n is a real constant. This is distinct from exponential functions where the variable appears in the exponent, such as 2ˣ.

幂函数的形式为 f(x) = xⁿ，其中 x 是变量，n 是实常数。这与指数函数不同：指数函数中变量出现在指数位置，如 2ˣ。

• n = 2: f(x) = x² (quadratic function).

  n = 2：f(x) = x²（二次函数）。

• n = ½: f(x) = √x (square root function).

  n = ½：f(x) = √x（平方根函数）。

• n = -1: f(x) = 1/x (reciprocal function).

  n = -1：f(x) = 1/x（反比例函数）。

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2. The Basic Rule: d/dx (xⁿ) = n xⁿ⁻¹ | 基本求导法则

For any real constant n, the derivative of xⁿ with respect to x is n xⁿ⁻¹. This rule is often called the power rule.

对任意实常数 n，xⁿ 关于 x 的导数为 n xⁿ⁻¹。这条法则常被称为“幂法则”。

d/dx (xⁿ) = n xⁿ⁻¹

This formula is central to differentiation. It applies for every real value of n, including positive, negative and fractional exponents.

这个公式是微分学的核心。它对所有实数 n 都成立，包括正指数、负指数和分数指数。

Function / 函数	Derivative / 导数
x³	3x²
x⁻²	-2x⁻³
x^½	½ x^(-½)

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3. Verification for Positive Integer n | 正整数 n 的验证

Let n be a positive integer. From the definition of the derivative, we write the limit of the difference quotient:

设 n 为正整数。由导数的定义，我们写出差商的极限：

f'(x) = lim_{h→0} [(x+h)ⁿ – xⁿ]/h

Expanding (x+h)ⁿ by the binomial theorem gives xⁿ + n xⁿ⁻¹h + … + hⁿ. After subtracting xⁿ and dividing by h, the first term is n xⁿ⁻¹ and all remaining terms contain a factor of h. Letting h → 0 leaves only n xⁿ⁻¹.

用二项式定理展开 (x+h)ⁿ 得到 xⁿ + n xⁿ⁻¹h + … + hⁿ。减去 xⁿ 后除以 h，第一项为 n xⁿ⁻¹，其余各项都含有因子 h。令 h → 0 后只剩下 n xⁿ⁻¹。

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4. Extension to Negative and Fractional Exponents | 向负指数与分数指数的推广

The power rule remains valid when n is negative or a fraction. For example, rewrite 1/x as x⁻¹, then differentiate: d/dx (x⁻¹) = -x⁻², which is -1/x².

当 n 为负数或分数时，幂法则仍然成立。例如，将 1/x 写为 x⁻¹，则 d/dx (x⁻¹) = -x⁻²，即 -1/x²。

For square roots, rewrite √x as x^½, then the derivative is ½ x^(-½), often written as 1/(2√x). This result can also be verified using the general binomial series or implicit differentiation.

对于平方根，把 √x 写成 x^½，则导数为 ½ x^(-½)，通常写作 1/(2√x)。该结果也可以用广义二项级数或隐函数求导来验证。

The formula even works for n = 0. Since x⁰ = 1 is constant, its derivative is 0, which agrees with 0·x⁻¹ = 0 wherever the expression is defined.

这个公式对 n = 0 也适用。由于 x⁰ = 1 是常数，其导数为 0，这与 0·x⁻¹ = 0 在其有定义处一致。

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5. Special Cases to Memorise | 需要记住的特殊情况

The following special cases appear frequently in exams and should be memorised:

下列特殊情况在考试中频繁出现，应当熟记：

• n = 1: f(x) = x, f'(x) = 1.

  n = 1：f(x) = x，f'(x) = 1。

• n = 0: f(x) = 1, f'(x) = 0.

  n = 0：f(x) = 1，f'(x) = 0。

• n = ½: f(x) = √x, f'(x) = 1/(2√x).

  n = ½：f(x) = √x，f'(x) = 1/(2√x)。

• n = -1: f(x) = 1/x, f'(x) = -1/x².

  n = -1：f(x) = 1/x，f'(x) = -1/x²。

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6. Finding Tangents and Normals | 求切线与法线

The derivative of a power function gives the slope of the tangent line at any point. For example, take y = x³. Then dy/dx = 3x², so at x = 1 the slope is 3.

幂函数的导数给出图像上任意一点处切线的斜率。例如，设 y = x³，则 dy/dx = 3x²，所以在 x = 1 处斜率为 3。

At the point (1, 1), the tangent equation is y – 1 = 3(x – 1), which simplifies to y = 3x – 2. Since the normal is perpendicular, its slope is the negative reciprocal -1/3, giving y – 1 = -1/3(x – 1), or y = -x/3 + 4/3.

在点 (1, 1) 处，切线方程为 y – 1 = 3(x – 1)，化简得 y = 3x – 2。由于法线与切线垂直，其斜率为倒数负数 -1/3，因此 y – 1 = -1/3(x – 1)，即 y = -x/3 + 4/3。

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7. Rates of Change and Motion | 变化率与运动学

In kinematics, if displacement is given by a power function s = tⁿ, then velocity is v = ds/dt = n tⁿ⁻¹ and acceleration is a = d²s/dt² = n(n-1)tⁿ⁻².

在运动学中，若位移由幂函数 s = tⁿ 给出，则速度 v = ds/dt = n tⁿ⁻¹，加速度 a = d²s/dt² = n(n-1)tⁿ⁻²。

For instance, if s = t³, then v = 3t² and a = 6t. These results follow immediately from applying the power rule twice.

例如，若 s = t³，则 v = 3t²，a = 6t。这些结果可以直接由幂法则应用两次得到。

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8. Curve Sketching and Monotonicity | 函数作图与单调性

The sign of f'(x) tells us whether a function is increasing or decreasing. For f(x) = x², f'(x) = 2x, so the function decreases for x < 0 and increases for x > 0.

f'(x) 的符号告诉我们函数是递增还是递减。对于 f(x) = x²，f'(x) = 2x，因此当 x < 0 时函数递减，当 x > 0 时函数递增。

For f(x) = x³, f'(x) = 3x² is non-negative for all x, so the function is always increasing and has a stationary point at x = 0. Such information is essential for sketching accurate graphs.

对于 f(x) = x³，f'(x) = 3x² 对所有 x 都非负，因此函数始终递增，并在 x = 0 处有驻点。这些信息对于准确作图至关重要。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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