Differentiation of Quadratic Functions | 二次函数的求导技巧

📚 Differentiation of Quadratic Functions | 二次函数的求导技巧

In A-Level Mathematics, differentiation is a fundamental operation in pure maths. For any quadratic function of the form f(x) = ax² + bx + c, the derivative is f'(x) = 2ax + b. This simple result gives the gradient of the curve at any point and unlocks the analysis of tangents, stationary points and curve sketching.

在 A-Level 数学中，求导是纯数部分的基本运算。对于任何形如 f(x) = ax² + bx + c 的二次函数，其导数为 f'(x) = 2ax + b。这个简洁的结果给出曲线在任意一点的斜率，并帮助我们研究切线、驻点以及函数图像。

1. Understanding the Derivative | 理解导数

The derivative measures the rate at which a function changes. Geometrically, f'(x) tells you the slope of the tangent line to the curve y = f(x) at the point (x, f(x)). For a straight line, the gradient is constant, but for a quadratic curve the gradient changes as x changes.

导数衡量的是函数变化的快慢。从几何上看，f'(x) 表示曲线 y = f(x) 在点 (x, f(x)) 处切线的斜率。对于直线，斜率是固定的；而对于二次曲线，斜率会随着 x 的变化而改变。

The formal definition you may have seen in the Edexcel Pure Mathematics specification is:

你在 Edexcel 纯数考试大纲中可能见过的正式定义为：

f'(x) = lim (h → 0) [f(x + h) – f(x)] / h

This limit, when it exists, is the gradient function, also called the derived function.

这个极限如果存在，就是梯度函数，也叫做导函数。

2. The Power Rule Applied to x² | 幂法则应用于 x²

The quickest way to differentiate a quadratic is to use the power rule: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. For n = 2, we obtain f'(x) = 2x.

对二次函数求导最快的方法是使用幂法则：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。当 n = 2 时，我们得到 f'(x) = 2x。

For the specific function f(x) = x², we can also prove this from first principles:

对于特定函数 f(x) = x²，我们也可以用导数的定义来证明：

f'(x) = lim (h → 0) [((x + h)² – x²) / h] = lim (h → 0) [(2xh + h²) / h] = lim (h → 0) [2x + h] = 2x

Since the final term h tends to 0, the gradient of y = x² at any point x is exactly 2x. For example, at x = 3, the gradient of y = x² is 6.

因为最后一项 h 趋于 0，所以 y = x² 在任意点 x 处的斜率正好是 2x。例如，当 x = 3 时，y = x² 的切线斜率为 6。

3. Differentiating ax² | 对 ax² 求导

When a quadratic term has a coefficient, we use the constant multiple rule. If f(x) = ax², then f'(x) = a × 2x = 2ax.

当二次项带有系数时，我们使用常数倍数法则。若 f(x) = ax²，则 f'(x) = a × 2x = 2ax。

Here are some quick examples:

下面是一些快速示例：

Notice that the original power of 2 is brought down and multiplied by the coefficient; after differentiating, the power decreases by 1.

注意，原来的幂 2 被提到前面并与系数相乘；求导后，幂指数

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