📚 Derivative of xn (n = 0, 1, 2, 3, … ) | 幂函数 xn 的导数 (n = 0, 1, 2, 3, … )
e of power functions of the form f(x) = xⁿ, where n is a non-negative integer (0, 1, 2, 3, … ency and a deeper understanding of how rates of change are computed from first principles.
微分是微积分中与积分并列的两大核心运算之一。IB 数学中的一项基本技能就是求形如 f(x) = xⁿ（n 为非负整数 0, 1, 2, 3, …）的幂函数的导数。这些函数在建模中无处不在，从简单的线性关系到高次多项式。掌握 xⁿ 的求导法则不仅能提升运算熟练度，还能加深对如何从第一原理计算变化率的理解。
1. Introduction to Differentiation and Power Functions | 导数与幂函数简介
a constant exponent, are the building blocks of polynomial functions. For n = 0, 1, 2, 3, … f(x) = xⁿ describes a curve whose gradient can be expressed neatly by a simple power rule.
在微积分中，函数 f 在点 x 处的导数表示该点的瞬时变化率或切线的斜率。幂函数，即变量乘以常数次幂的函数，是多项式函数的基石。当 n = 0, 1, 2, 3, … 时，函数 f(x) = xⁿ 描述的曲线，其梯度可以用一个简洁的幂法则来表达。
2. The First Principle of Differentiation | 导数的第一原理
The derivative of a function f(x) is defined by the limit: f' vided the limit exists. This first principle will be used to derive the derivative of xⁿ for small integer values of n, before we generalise the result.
函数 f(x) 的导数由以下极限定义：f'(x) = lim (h→0) [f(x+h) - f(x)] / h，假设该极限存在。我们将用这一第一原理推导 n 为小整数值时 xⁿ 的导数，然后再推广结果。
3. Derivative of a Constant Function (n = 0) | 常数函数的导数 (n = 0)
= 1 (assuming x ≠ 0). Using the first principle, f(x+h) = 1, so f(x+h) - f(x) = 0. Thus f' , confirming that the derivative of any constant is 0. This matches the rule d/dx (c) = 0.
当 n = 0 时，f(x) = x⁰ = 1（假设 x ≠ 0）。利用第一原理，f(x+h) = 1，故 f(x+h) - f(x) = 0。因此 f'(x) = lim (h→0) 0/h = 0。水平线的梯度为零，印证了常数的导数为 0，这与 d/dx (c) = 0 的法则一致。
4. Derivative of f(x) = x (n = 1) | f(x) = x 的导数 (n = 1)
The difference quotient is [(x+h) - x] / h = h / h = 1. Taking the limit as h → 0 gives f'(x) = 1. This tells us the line y = x has a constant slope of 1 everywhere.
设 f(x) = x，则 f(x+h) = x + h。差商为 [(x+h) - x] / h = h / h = 1。取 h → 0 时的极限，得 f'(x) = 1。这说明直线 y = x 处处斜率为 1。
5. Derivative of f(x) = x² (n = 2) | f(x) = x² 的导数 (n = 2)
+ 2xh + h² - x²] / h = (2xh + h²) / h = 2x + h. As h → 0, the term h vanishes, yielding f'(x) = 2x. This shows the gradient of y = x² grows linearly with x.
对于 f(x) = x²，有 f(x+h) = (x+h)² = x² + 2xh + h²。差商为 [x² + 2xh + h² - x²] / h = (2xh + h²) / h = 2x + h。当 h → 0 时，项 h 消失，得到 f'(x) = 2x。这表明 y = x² 的梯度随 x 线性增长。
6. Derivative of f(x) = x³ (n = 3) | f(x) = x³ 的导数 (n = 3)
dividing by h gives (3x²h + 3xh² + h³)/h = 3x² + 3xh + h². In the limit h → 0, we obtain f' e derivative of xⁿ brings the exponent down as a coefficient and reduces the power by one.
展开 (x+h)³ = x³ + 3x²h + 3xh² + h³。减去 x³ 并除以 h 得 (3x²h + 3xh² + h³)/h = 3x² + 3xh + h²。在 h → 0 的极限下，得到 f'(x) = 3x²。注意规律：xⁿ 的导数将指数下移为系数，并将幂次减 1。
7. The General Rule for xⁿ (n ∈ ℕ) | xⁿ 的一般导数法则 (n 为自然数)
1, 2, 3 we conjecture a general formula. For any positive integer n, if f(x) = xⁿ, then f' ⁻¹ is taken as 0 (with the understanding that the rule applies for x > 0 where necessary).
由 n = 1, 2, 3 的特例我们猜想一个通用公式。对于任何正整数 n，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。这被称为幂法则。当 n = 0 时也成立，只要我们将 x⁰ = 1，并注意到 0 · x⁻¹ 被当作 0（且在必要处限定 x > 0）。
If f(x) = xⁿ, then f'(x) = n xⁿ⁻¹, for n = 0, 1, 2, 3, …
若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n = 0, 1, 2, 3, …
8. Proof Using Binomial Theorem | 利用二项式定理证明
, write (x+h)ⁿ using the binomial expansion: (x+h)ⁿ = xⁿ + n xⁿ⁻¹ h + [n(n-1)/2] xⁿ⁻² h² + … + hⁿ. Subtract xⁿ and divide by h:
为证明对任意正整数 n 的法则，写出 (x+h)ⁿ 的二项式展开： (x+h)ⁿ = xⁿ + n xⁿ⁻¹ h + [n(n-1)/2] xⁿ⁻² h² + … + hⁿ。减去 xⁿ 并除以 h：
[ (x+h)ⁿ - xⁿ ] / h = n xⁿ⁻¹ + [n(n-1)/2] xⁿ⁻² h + … + hⁿ⁻¹
east one factor of h. As h → 0, all those terms tend to zero, leaving only n xⁿ⁻¹. Hence f'(x) = n xⁿ⁻¹. The proof confirms the pattern observed earlier and is valid for all n ∈ ℕ.
第一项之后的每一项都至少含有一个因子 h。当 h → 0 时，所有这些项都趋向于零，仅留下 n xⁿ⁻¹。因此 f'(x) = n xⁿ⁻¹。该证明确认了之前观察到的规律，并对所有 n ∈ ℕ 有效。
9. Worked Examples | 计算示例
Applying the power rule is straightforward. Here are a few examples to build confidence:
应用幂法则非常直接。以下示例有助建立信心：
If f(x) = x⁵, then f'(x) = 5x⁴.
若 f(x) = x⁵，则 f'(x) = 5x⁴。
If g(x) = x¹⁰, then g'(x) = 10x⁹.
若 g(x) = x¹⁰，则 g'(x) = 10x⁹。
If h(x) = x⁷³, then h'(x) = 73x⁷².
若 h(x) = x⁷³，则 h'(x) = 73x⁷²。
ative of 4x³ is 4 · 3x² = 12x². In general, d/dx [k · xⁿ] = k · n xⁿ⁻¹ for any constant k.
该法则可与常数倍结合。例如，4x³ 的导数为 4 · 3x² = 12x²。一般地，对于任意常数 k，有 d/dx [k · xⁿ] = k · n xⁿ⁻¹。
10. Higher-Order Derivatives | 高阶导数
can differentiate repeatedly to obtain higher-order derivatives. The second derivative, f''(x), is the derivative of f'(x). For f(x) = xⁿ, we have f'(x) = n xⁿ⁻¹, so f''(x) = n(n-1) xⁿ⁻² (for n ≥ 2). Continuing, the m-th derivative (m ≤ n) is:
由于 xⁿ 的导数仍是幂函数，我们可以反复求导得到高阶导数。二阶导数 f''(x) 即为 f'(x) 的导数。对于 f(x) = xⁿ，有 f'(x) = n xⁿ⁻¹，于是 f''(x) = n(n-1) xⁿ⁻²（n ≥ 2）。继续下去，第 m 阶导数（m ≤ n）为：
f(m)(x) = n(n-1)(n-2)…(n-m+1) xⁿ⁻ᵐ
For m > n, the derivative becomes zero. This factorial-like behaviour is important in polynomial calculus and Taylor series.
当 m > n 时，导数变成零。这种类似阶乘的行为在多项式微积分和泰勒级数中非常重要。
11. Equations of Tangent Lines | 切线方程
n find the equation of the tangent line to y = xⁿ at a point (a, aⁿ). The slope there is f'(a) = n aⁿ⁻¹, and the tangent line has equation y - aⁿ = n aⁿ⁻¹ (x - a).
一旦求出导数，就能找到曲线 y = xⁿ 在点 (a, aⁿ) 处的切线方程。该点斜率为 f'(a) = n aⁿ⁻¹，切线方程为 y - aⁿ = n aⁿ⁻¹ (x - a)。
1. The point is (1, 1), slope = 4·1³ = 4. Tangent equation: y - 1 = 4(x - 1) → y = 4x - 3.
示例：求 y = x⁴ 在 x = 1 处的切线。这里 n = 4, a = 1。点为 (1, 1)，斜率 = 4·1³ = 4。切线方程：y - 1 = 4(x - 1) → y = 4x - 3。
12. Summary and Further Extensions | 总结与拓展
We have shown that the derivative of xⁿ for n = 0, 1, 2, 3, … hematics, students will later see that the same formula holds for any real exponent n (with x > 0), and will learn to handle sums, products, quotients and composite functions. The simple power rule is your first and most versatile differentiation tool.
我们已经证明，对于 n = 0, 1, 2, 3, …，xⁿ 的导数为 n xⁿ⁻¹。这一通过第一原理和二项式定理得以验证的法则，是求所有多项式导数的基础。在 IB 数学中，学生之后会看到相同的公式对任意实数指数 n（x > 0）也成立，并将学习处理和、积、商和复合函数。简单的幂法则是你第一个且最通用的求导工具。
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