How to find a general expression for derivative

[east]

Hello everybody,

I am trying to solve this problem but I can't find an answer reading my book :(

Let z=F(u,w,x) where u=f(x,y) and w=g(x,y)

(i) find a general expression for D_xz
(ii) verify your answer in the case where F(u,w,x)=w³-2ux, f(x,y)=4xy-x+1 and g(x,y)=y+xy²

Many thanks

 

[Ray Vickson]

Hello everybody,

I am trying to solve this problem but I can't find an answer reading my book :(

Let z=F(u,w,x) where u=f(x,y) and w=g(x,y)

(i) find a general expression for D_xz
(ii) verify your answer in the case where F(u,w,x)=w³-2ux, f(x,y)=4xy-x+1 and g(x,y)=y+xy²

Many thanks

Other than trying to find it in your book, what else have you tried? Forum rules require you to show your work.

RGV

 

[east]

Yes, I've read the rules, sorry. the point is that I don't know where to stard with :( more than the just the solution I would like some references to books or links where I can study the problem..

thx again

 

[Ray Vickson]

Yes, I've read the rules, sorry. the point is that I don't know where to stard with :( more than the just the solution I would like some references to books or links where I can study the problem..

thx again

So, are you saying you have never encountered such material before? You have never heard of the chain rule?

RGV

 

[east]

sadly no :( I have to take a test of basic math knowledge and we only have some notes to prepare... but ty for the hint of the chain rule :)

 

[east]

ok, thanks to RVG who explained me where to look for the solution I think I have solved it:

(i): Dz/D_x=DF/D_u*DU/D_x+DF/D_w*DW/D_x+DF/D_x

(ii) Dz/D_x= -2x(4y-1)+3w²y²-2u

Is it right? it is fairly easy once discovered which formula to use (if I've used the right one of course)

 

[Ray Vickson]

ok, thanks to RVG who explained me where to look for the solution I think I have solved it:

(i): Dz/D_x=DF/D_u*DU/D_x+DF/D_w*DW/D_x+DF/D_x

(ii) Dz/D_x= -2x(4y-1)+3w²y²-2u

Is it right? it is fairly easy once discovered which formula to use (if I've used the right one of course)

Yes, you have made a correct start in part (i) and you have a correct result in part (ii) (although I guess there is an issue of whether or not to plug in the actual values of u and w and do a complete simplification).

RGV

 

[SammyS]

ok, thanks to RVG who explained me where to look for the solution I think I have solved it:

(i): Dz/D_x=DF/D_u*DU/D_x+DF/D_w*DW/D_x+DF/D_x

(ii) Dz/D_x= -2x(4y-1)+3w²y²-2u

Is it right? it is fairly easy once discovered which formula to use (if I've used the right one of course)

Yes, that looks good.

I used WolframAlpha to compare:

Your answer after substituting for w & u.

First substituting for w & u in F(u, w, x) then getting the partial derivative of that expression with respect to x .​