Alevel數學干貨，微積分之鏈式法則

Alevel數學干貨，微積分之鏈式法則

1.The China rule (鏈式法則)

If y=f(u) is a differentiable function of u and u=g(x) is a differentiable function of x, then y=f(g(x)) is a differentiable function of x and

or, equivalently,

如果復合函數處處可導，可使用鏈式法則來進行求導，外部函數的導數乘以內部函數的導數。

常考題型解析：

Example 1

SOLUTION

As you saw earlier, you can break down this expression as follows.

Differentiation these gives

and

By the chain rule

2. Related-Rate Problems

相關變化率問題

Differentiation with respect to different variables

對于不同變量的微分

The chain rule makes it possible to differentiate with respect to a variable which does not feature in the original expression. For example, the volume V of a sphere of radius r is given by

Differentiating this with respect to r gives the rate of change of volume with radius,

However you might be more interested in finding dv/dt, the rate of change of change of volume with time, t. To find this, you would use the chain rule:

相關變化率問題是復合函數求導的應用，例,半徑為r的球體積為

體積對半徑進行微分，可得到體積對于半徑的變化率，

要求出體積對于時間的變化率dV/dt，可以使用鏈式法則，通過半徑r對時間t進行微分。

常考題型解析

Example 2