干货满满｜Alevel数学微积分之链式法则 　　

 　Alevel数学干货，微积分之链式法则

　　1.The China rule (链式法则)

　　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,

　　如果复合函数处处可导，可使用链式法则来进行求导，外部函数的导数乘以内部函数的导数。

　　常考题型解析：

　　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

　　相关变化率问题

　　Differentiation with respect to different variables

　　对于不同变量的微分

　　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:

　　相关变化率问题是复合函数求导的应用，例,半径为r的球体积为

　　体积对半径进行微分，可得到体积对于半径的变化率，

　　要求出体积对于时间的变化率dV/dt，可以使用链式法则，通过半径r对时间t进行微分。

　　常考题型解析

　　Example 2

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