Differentiating Polynomials: Power of at Least One Term Decreased by 1 | 多项式求导：至少一项的幂指数降低 1

📚 Differentiating Polynomials: Power of at Least One Term Decreased by 1 | 多项式求导：至少一项的幂指数降低 1

For Edexcel A-Level Mathematics, the power rule is the first and most important differentiation technique. When you differentiate a term such as xⁿ, the exponent n comes down as a multiplier, and the new power is n − 1. In other words, the power of at least one term is decreased by 1 in every differentiation step.

在 Edexcel A-Level 数学中，幂法则是最先也是最重要的求导方法。当你对 xⁿ 这样的项求导时，指数 n 会作为乘数移到前面，而新的指数变为 n − 1。换句话说，在每一步求导中，至少有一项的幂指数会降低 1。

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1. The Power Rule | 幂法则

If f(x) = xⁿ, where n is any real constant, then the derivative is f'(x) = n xⁿ⁻¹. This rule is fundamental for differentiating polynomials because it turns every power of x into a new term with a lower power.

如果 f(x) = xⁿ，其中 n 是任意实常数，那么它的导数就是 f'(x) = n xⁿ⁻¹。这个法则是多项式求导的基础，因为它把每一个 x 的幂转化为一个新的项，并且新的幂比原来低。

d/dx (xⁿ) = n xⁿ⁻¹

For example, the derivative of x⁵ is 5x⁴. The original power 5 has been reduced by 1 to give 4. The derivative of x² is 2x¹, which is simply written as 2x.

例如，x⁵ 的导数是 5x⁴。原来的幂 5 减去了 1，得到新的幂 4。x² 的导数是 2x¹，通常简写为 2x。

Original function f(x)	Derivative f'(x)
x²	2x
x³	3x²
x⁴	4x³

Notice that in every row, the new exponent is exactly one less than the original exponent. This is the key pattern tested in Edexcel exams.

注意在每一行中，新的指数恰好比原来的指数少 1。这是 Edexcel 考试中考查的关键规律。

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2. Why ‘Power Decreased by 1’ Matters | 为什么“幂指数降低 1”很重要

Differentiation measures the rate of change of a function. For a power function xⁿ, lowering the power by 1 reflects the fact that the gradient grows more slowly than the original function. It also ensures the derivative of a linear term x¹ becomes the constant x⁰ = 1.

求导测量的是函数的变化率。对于幂函数 xⁿ，幂指数降低 1 反映了这样一个事实：梯度的增长比原函数更慢。这也确保了线性项 x¹ 的导数变成常数 x⁰ = 1。

This pattern helps you predict the degree of the derivative: differentiating a polynomial of degree n always gives a polynomial of degree n − 1, provided the leading term is not constant. For example, a cubic polynomial becomes a quadratic after differentiation.

这个规律可以帮助你预测导数的次数：对一个 n 次多项式求导，得到的多项式次数总是 n − 1，前提是首项不是常数。例如，三次多项式求导后变成二次多项式。

Geometrically, the power decreasing by 1 means the gradient function has a lower degree than the original curve. This is why the gradient of a quadratic is a straight line, and the gradient of a cubic is a quadratic curve.

从几何角度看，幂指数降低 1 意味着梯度函数的次数比原曲线低。这就是为什么二次曲线的梯度是一条直线，三次曲线的梯度是一条二次曲线。

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3. Applying to Polynomials Term by Term | 逐项应用于多项式

You can differentiate a polynomial by applying the power rule to each term separately. The derivative of a sum is the sum of the derivatives, so you can work through the expression from left to right.

你可以对多项式中的每一项分别应用幂法则来求导。和的导数等于导数的和，因此你可以从左到右逐项求导。

For f(x) = 3x⁴ − 5x³ + 2x² − 7x + 9, the derivative is f'(x) = 12x³ − 15x² + 4x − 7. Notice each non-constant term has its power lowered by 1, while the constant 9 disappears because the derivative of any constant is 0.

对于 f(x) = 3x⁴ − 5x³ + 2x² − 7x + 9，其导数为 f'(x) = 12x³ − 15x² + 4x − 7。注意每个非常数项的幂都降低了 1，而常数 9 消失了，因为任何常数的导数都是 0。

Another example: if g(x) = 2x⁶ + x³ − 4x, then g'(x) = 12x⁵ + 3x² − 4. The term 4x has power 1, so differentiating gives 4x⁰ = 4, and the sign is kept from the original expression.

另一个例子：如果 g(x) = 2x⁶ + x³ − 4x，那么 g'(x) = 12x⁵ + 3x² − 4。4x 这一项的幂是 1，求导后得到 4x⁰ = 4，同时保留原表达式中的符号。

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4. Constant Multiples and Sums | 常数倍与和差法则

If a term has a constant coefficient, keep the coefficient and differentiate the power part. For example, d/dx (7x⁶) = 7 × 6x⁵ = 42x⁵. The coefficient 7 is not affected by the power change.

如果某一项有常数系数，求导时保留该系数，再对幂部分求导。例如，d/dx (7x⁶) = 7 × 6x⁵ = 42x⁵。系数 7 不会受到幂变化的影响。

If two functions are added or subtracted, differentiate each one separately. This is why polynomials are straightforward to differentiate term by term; there is no need for the product rule unless two non-constant expressions are multiplied together.

如果两个函数相加或相减，就分别对每个函数求导。这就是为什么多项式可以逐项轻松求导的原因；除非两个非常数表达式相乘，否则不需要使用乘法法则。

In general, for constants a and b, d/dx [a f(x) + b g(x)] = a f'(x) + b g'(x). This linearity is essential and is often used without being explicitly stated in exam solutions.

一般地，对于常数 a 和 b，d/dx [a f(x) + b g(x)] = a f'(x) + b g'(x)。这种线性性质至关重要，在考试答案中经常被直接使用而不加以说明。

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5. Negative Powers | 负指数

The power rule also works when n is negative. For example, if f(x) = x⁻², then f'(x) = −2x⁻³. The power has been decreased by 1 from −2 to −3, so the result is more negative.

当 n 为负数时，幂法则同样适用。例如，如果 f(x) = x⁻²，那么 f'(x) = −2x⁻³。幂从 −2 降低到 −3，减少了 1，因此结果负得更多。

This is particularly useful for differentiating expressions such as 1/x or 1/x², which can be rewritten as x⁻¹ and x⁻² before applying the rule. For f(x) = 1/x = x⁻¹, the derivative is f'(x) = −1x⁻² = −1/x².

这对于求 1/x 或 1/x² 这类表达式的导数特别有用，它们可以分别先改写为 x⁻¹ 和 x⁻²，然后再应用幂法则。对于 f(x) = 1/x = x⁻¹，其导数为 f'(x) = −1x⁻² = −1/x²。

Always rewrite reciprocals as negative powers before differentiating. This avoids errors and makes the power-decrease step visible.

求导前一定要把倒数改写为负指数形式。这样可以避免错误，并且让幂指数降低的步骤清晰可见。

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6. Fractional Powers | 分数指数

Fractional powers such as x^½ or x^⅓ can also be differentiated using the same rule. For f(x) = x^½, the derivative is f'(x) = ½ x^−½. The power ½ has been decreased by 1

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