Advanced partial differentiation

[isa.b]

Homework Statement

Given two functions F and G, I will use the following notation to indicate partial differentiation:
F_x means dF/dx
G_z means dG/dz
(for example)

I would like to develop the following two expressions. I don't want them grouped into brackets as they're now, but I have to develop them into single terms using the rules of partial differentiation.

Homework Equations

(F_x² + F_y²)_x = ?

(F_xG_x + F_yG_y)_y = ?

The Attempt at a Solution

Your help is very much appreciated, thanks in advance!

 

[hunt_mat]

What have you done so far? What would you do if they were ordinary derivatives for example?

(f'^2+g'^2)'=?

 

[isa.b]

What have you done so far? What would you do if they were ordinary derivatives for example?

(f'^2+g'^2)'=?

I would break them into two separate derivatives
(f'^2)' and same for g

then 2f'

but I wonder what happens with a different axis such as (Fy^2)x
even worse with the products (the second case)

unfortunately I don't have much theoretical knowledge on partial differentiation (it's been many years ago...)

 

[HallsofIvy]

There is nothing at all advanced about those, and no "theoretical knowledge" is need. You just use the basic rules for differentiation.

$$(F_x+G_x + F_yG_y)_y = (F_xG_y)_y+ (F_xG_y)_y$$
$$= F_{xx}G_y+ F_xG_{xy}+ F_{xy}G_y+ F_xG_{yy}$$

 

[isa.b]

There is nothing at all advanced about those, and no "theoretical knowledge" is need. You just use the basic rules for differentiation.

$$(F_x+G_x + F_yG_y)_y = (F_xG_y)_y+ (F_xG_y)_y$$
$$= F_{xx}G_y+ F_xG_{xy}+ F_{xy}G_y+ F_xG_{yy}$$

Thanks for your help HallsofIvy.
I'm not a mathematician, so I find it quite difficult, please understand.

The expression you worked out is for my 2nd case right? (I guess you just did a couple of typos, but that's why I'm not sure).
With your example I think I should be able to work out also the 1st case, but do you mind to show me that too? I'd like to check if I actually can do it

Thanks again!