Calculus Question within Lagrangian mechanics

[Hennessy]

TL;DR Summary

Product rules with hidden chain rules

Hi all currently got a lagrangian function which i've found to be :
\begin{equation}\mathcal{L}=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+4x^2\dot{x}^2+4y^2\dot{y}^2+8xy\dot{x}\dot{y})- mg(x^2+y^2)
\end{equation}
Let us first calculate
$$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to $$m\dot{x}+4x^2m\dot{x}+8xy\dot{y}$$ now we also have to differentiate this again with respect to t.
$$\frac{d}{dt}(m\dot{x}+4x^2m\dot{x}+8xy\dot{y}) $$ Now this is where i'm stuck. I'm stuck because of the 2 product rule in the middle of this term and then the triple product rule on the right hand and then within them i know there are chain rules as $$x,y,\dot{x},\dot{y}$$ are all f(t). Im basically asking how to use the product rule again. using $$x,\dot{x}$$ as my uv then the product rule is $$x'\dot{x}+x\dot{x}'$$ but when i calculate the primes i get confused. so for example $$x'\dot{x}+x\dot{x}'$$ does this mean differentiate the entire function wrt x and then multiply it just by $\dot{x}$ or does it mean multiply it by the entire thing? Advice would be appreciated , i know this is more a calculus question but just trying to figure it out apologies if this is in the wrong place. Put it here as it requires knowledge of lagrangian mechanics for $$x,y,\dot{x},\dot{y}$$ as being time derivatives is all. Thank you!

 

[Orodruin]

Use the chain rule for derivatives. If you have a function $F(t,f_1(t),f_2(t),\ldots)$ then
$$
\frac{dF}{dt} =
\frac{\partial F}{\partial t} + \frac{\partial F}{\partial f_1} \dot f_1 + \frac{\partial F}{\partial f_2} \dot f_2 + \ldots
$$

You may want to redo this derivative:

Let us first calculate
$$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to $$m\dot{x}+4x^2m\dot{x}+8xy\dot{y}$$

It is also unclear what you mean by $x’$. We already use dots to denote time derivatives.

 

[Hennessy]

Use the chain rule for derivatives. If you have a function $F(t,f_1(t),f_2(t),\ldots)$ then
$$
\frac{dF}{dt} =
\frac{\partial F}{\partial t} + \frac{\partial F}{\partial f_1} \dot f_1 + \frac{\partial F}{\partial f_2} \dot f_2 + \ldots
$$

You may want to redo this derivative:


It is also unclear what you mean by $x’$. We already use dots to denote time derivatives.

Hi there, apologies for the confusion. I rewrote the product rule from uv' +vu' and I wrote it in terms of my two functions x and $\dot{x}$ i also understand that i originally wrote my product rule as u'v+v'u , but this shouldn't of changed the result if im not mistaken?