# Chain rule

1. Feb 22, 2009

### -EquinoX-

1. The problem statement, all variables and given/known data

Let $$F(u,v)$$ be a function of two variables. Find f '(x) for $$f(x) = F(x, 6)$$.

2. Relevant equations

3. The attempt at a solution

I need to find the answer in terms of $$F_u$$, how can I do this?

2. Feb 22, 2009

### Dick

Remember what the definition of F_u is.

3. Feb 22, 2009

### -EquinoX-

I am confused what are we supposed to find the derivative of F_u with respect to what? x or y?

4. Feb 22, 2009

### Dick

F_u is the derivative of F(u,v) with respect to the first variable with the second variable held constant, right? f'(x) looks like pretty much the same thing.

5. Feb 22, 2009

### -EquinoX-

my guess is that it will be something like:

F_u * v(6)

is this right?

6. Feb 22, 2009

### Dick

What does that mean??? Try an example f(u,v)=u^2-u*v. What's F_u? What is f'(x)?

7. Feb 22, 2009

### -EquinoX-

I assume that F_u is the derivative of f(u,v)

f(u,v)=u^2-u*v

is just:

(u -v) is this correct?

8. Feb 22, 2009

### Staff: Mentor

No, $F_u$ is the partial derivative of F(u, v). You calculate it by assuming that v is constant and that only u is changing. $F_u$ can also be written as $\partial F/ \partial u$
No.
One way to to calculate this is to take the limit:
$$\lim_{h \rightarrow 0} \frac{F(u + h, v) - F(u, v)}{h}$$

9. Feb 22, 2009

### Dick

You mean F_u(u,v)=2u-v, yes? Now what's are f(x) and f'(x)?

10. Feb 22, 2009

### -EquinoX-

f(x) is therefore 2x-6 right? and f'(x) is 2 ?

11. Feb 22, 2009

### Staff: Mentor

Equinox,
Don't forget that you're answering Dick's question about a specific example (post 6), not the question you first posted. He's trying to get you to think about this the right way.

12. Feb 22, 2009

### -EquinoX-

Yes I am aware of that.. I am supposed to find the relation between the example he's given and the real answer to my question. I believe so the answer is then F_u(x,6) ?

13. Feb 22, 2009

### Dick

If you mean f'(x)=F_u(x,6), yes. f(x)=2x-6, and f'(x)=2 are NOT right.

14. Feb 22, 2009

### -EquinoX-

so f'(x)=F_u(x,6) is not correct?

15. Feb 22, 2009

### Dick

Yes it is. The other two things aren't.

16. Feb 22, 2009

### -EquinoX-

what if a variation of the question asks for f(x) = F(x, x), is it just then 0?

17. Feb 22, 2009

### Dick

No, why would you say that? Wouldn't you need to think about F_v as well? v isn't fixed anymore.

18. Feb 22, 2009

### -EquinoX-

Hmm..is x here actually a number or is x another function?

19. Feb 22, 2009

### Dick

Actually, sorry, I meant yes. For the specific function F(u,v)=u^2-u*v, sure f(x)=F(x,x)=0. f'(x)=0. Sorry, I misspoke. But you can't express f'(x) purely in terms of F_u.

20. Feb 22, 2009

### -EquinoX-

hmm.. I tried to input 0 as the answer and it didn't accept it..