hi there, i am actually studying about functional equation.

I got stucked with some derivatives problem,

and where i could find nowhere to refer or study from,

because it seems it is out of university book level.

my question is this :

what does it means by taking derivative with respect to partial derivative?

can anyone visualize this idea to me?

because i couldn't figure out the term with respect to partial derivative,

when it comes to a functional equation,

of which is a differential equations.

For e.g : F(x,y(x),y'(x),y''(x)) , find \(\displaystyle \frac{d}{dx}\) of \(\displaystyle \partial \) F(x,y(x),y'(x),y''(x)) / \(\displaystyle \partial \) y' .

How can we write the full solution with partial derivative respect to y' ? and how bout y''?

I assume you're talking about a standard Calculus of Variations problem with a functional of the form \(\displaystyle J=\int_a^b F(x,y,y',y'')dx\). Now, for this specific case the variational derivative is of the form \(\displaystyle F_y-\frac{\partial }{\partial x}F_{y'}=0\). Now, I agree this can be confusing at first but what that really means is call \(\displaystyle F(x,y,y',y'')=F(t,u,v,w)\). So, for example if \(\displaystyle F(x,y,y',y'')=x^2y+y'+\frac{y'}{y''}\) you would get \(\displaystyle F(t,u,v,w)=t^2u+v+\frac{v}{w}\). Now, the Euler-Lagrange equation then says take \(\displaystyle F_u\) so in this case \(\displaystyle F_u=t^2\) and \(\displaystyle F_v=1+\frac{1}{v}\). But! You have to sub back in what that really means, i.e. \(\displaystyle F_y=x^2,F_{y'}=1+\frac{1}{y''}\). Now, treating these again as actual functions of \(\displaystyle x\) you compute \(\displaystyle F_y-\frac{d}{dx}F_{y'}\). The point is that \(\displaystyle F_y,F_{y'}\) intend you to treat \(\displaystyle F\) as a function of the

*independent* variables \(\displaystyle x,y,y',y''\) and differentiate with respect to the specified one. But! Once done with that the \(\displaystyle \frac{d}{dx}\) intends you to treat \(\displaystyle F_y,F_{y'}\) again as a function of functions of \(\displaystyle x\).

Write back if you need more clatification.