Basic Differentiation Rules | 基本求导法则

📚 Basic Differentiation Rules | 基本求导法则

Differentiation is one of the cornerstones of calculus. In IB Mathematics, a secure grasp of basic differentiation rules allows you to analyse rates of change, gradients of curves, and optimisation problems with confidence. This guide explains each rule step by step, with exam-style examples.

求导是微积分的基石之一。在 IB 数学中，扎实掌握基本求导法则，能让你自信地分析变化率、曲线斜率与最优化问题。本指南逐步讲解每一条法则，并提供贴近考试的例题。

1. What is Differentiation? | 什么是求导？

Differentiation measures how a function changes as its input changes. For a function y = f(x), the derivative f′(x) represents the instantaneous rate of change, and geometrically it is the slope of the tangent line at a given point.

求导衡量的是函数随自变量变化的速率。对于函数 y = f(x)，导数 f′(x) 表示瞬时变化率，在几何上就是曲线上某一点处切线的斜率。

The formal definition from first principles is:

由极限定义，导数的形式化定义如下：

f′(x) = lim(Δx→0) [f(x + Δx) − f(x)] / Δx

Here Δx is a small change in x, and the limit captures the rate of change at a single point rather than over an interval.

其中 Δx 是 x 的一个微小变化量，极限帮助我们捕捉函数在某一点处的变化率，而不是一个区间上的平均变化率。

2. The Gradient of a Tangent | 切线的斜率

A secant line passes through two points on a curve, and its gradient is the average rate of change between those points. As the two points move closer together, the secant line approaches the tangent line.

割线经过曲线上两点，其斜率表示这两点之间的平均变化率。当两点逐渐靠近时，割线会趋近于切线。

At x = a, the value f′(a) gives the gradient of the tangent to the curve at the point (a, f(a)). This is why differentiating is often described as finding the slope of a curve at a point.

在 x = a 处，f′(a) 给出了曲线在点 (a, f(a)) 处切线的斜率。因此，求导常被描述为求曲线在某一点的斜率。

3. The Power Rule | 幂法则

For any real number n, the power rule states that the derivative of xⁿ is found by bringing the power to the front and reducing the power by one.

对于任意实数 n，幂法则指出：求 xⁿ 的导数时，把指数 n 提到前面，并将指数减 1。

d/dx (xⁿ) = n xⁿ⁻¹

For example, d/dx(x⁴) = 4x³ and d/dx(x²) = 2x. The power rule also works for negative powers, such as d/dx(x⁻¹) = −x⁻² = −1/x².

例如：d/dx(x⁴) = 4x³，d/dx(x²) = 2x。幂

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