数学知识复习：二阶导复合函数的链式法则

Posted 2022-02-10 UQI-LIUWJ

tags:

篇首语：本文由小常识网(cha138.com)小编为大家整理，主要介绍了数学知识复习：二阶导复合函数的链式法则相关的知识，希望对你有一定的参考价值。

1 结论

$\frac{d^2y}{dx^2}=\frac{d^2y}{du^2}(\frac{du}{dx})^2+\frac{d^2u}{dx^2}\frac{dy}{du}$

令 $z=\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

那么 $\frac{d^2y}{dx^2}=\frac{d(\frac{dy}{dx})}{dx}=\frac{dz}{dx}$

$=\frac{dz}{du}\frac{du}{dx}=\frac{d(\frac{dy}{du}\frac{du}{dx})}{du}\frac{du}{dx}$

$=(\frac{d(\frac{dy}{du})}{du}\frac{du}{dx} +\frac{d(\frac{du}{dx})}{du}\frac{dy}{du}) \frac{du}{dx}$

$=[(\frac{d^2y}{du^2})\frac{du}{dx} +\frac{d(\frac{du}{dx})}{du}\frac{dy}{du}]\frac{du}{dx}$

$=[(\frac{d^2y}{du^2})(\frac{du}{dx})^2] +[\frac{d(\frac{du}{dx})}{du}\frac{du}{dx}]\frac{dy}{du}$

$=[(\frac{d^2y}{du^2})(\frac{du}{dx})^2] +[\frac{d(\frac{du}{dx})}{dx}]\frac{dy}{du}$

$=\frac{d^2y}{du^2}(\frac{du}{dx})^2+\frac{d^2u}{dx^2}\frac{dy}{du}$

以上是关于数学知识复习：二阶导复合函数的链式法则的主要内容，如果未能解决你的问题，请参考以下文章