The Quotient Rule: Differentiating Functions in Division | 商法则：两个函数相除的求导

📚 The Quotient Rule: Differentiating Functions in Division | 商法则：两个函数相除的求导

When two differentiable functions are written as a quotient, the derivative is not simply the quotient of the two derivatives. The quotient rule provides a reliable formula for differentiating expressions of the form f(x)/g(x).

当两个可导函数写成商的形式时，其导数并不是简单地把两个导数相除。商法则为我们求形如 f(x)/g(x) 的表达式提供了可靠公式。

1. Definition and Formula | 定义与公式

Suppose h(x) = f(x)/g(x), where f and g are differentiable functions and g(x) ≠ 0. Then the quotient rule states:

设 h(x) = f(x)/g(x)，其中 f 和 g 都是可导函数，且 g(x) ≠ 0。那么商法则给出：

d/dx [ f(x) / g(x) ] = [ g(x) f'(x) − f(x) g'(x) ] / [ g(x) ]²

In particular, the derivative of a quotient is not the quotient of the individual derivatives. The rule must be applied whenever the denominator is not a constant.

特别要注意，f/g 的导数并不是 f’/g’。只要分母不是常数，就必须使用商法则。

2. Derivation from Product Rule | 从乘积法则推导商法则

The quotient rule can be derived from the product rule. Start with the equation f(x) = h(x)g(x), where h(x) = f(x)/g(x). Differentiating both sides with respect to x gives:

商法则可以由乘积法则推导出来。从等式 f(x) = h(x)g(x) 出发，其中 h(x) = f(x)/g(x)。两边对 x 求导得到：

f'(x) = h'(x)g(x) + h(x)g