## Multi-variable differential question

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

F = 1/2.a(T-Tc)M^2 + 1/4.bM^4

I need to find dF/dM

a,Tc,b are positive constants

2. The attempt at a solution

I assume this is to do with partial derivatives etc.

So I found:

∂F/∂T = 1/2.aM^2

∂F/∂M = TaM - TcaM + bM^3

And using a chain rule:

dF/dM = ∂F/∂T.dT/dM + ∂F/∂M.dM/dM

But not sure how to find dT/dM
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 Quote by steejk 1. The problem statement, all variables and given/known data F = 1/2.a(T-Tc)M^2 + 1/4.bM^4 I need to find dF/dM
Is that supposed to be$$F=(\frac 1 2)a(T-T_c)M^2 +(\frac 1 4)bM^4$$
 a,Tc,b are positive constants 2. The attempt at a solution I assume this is to do with partial derivatives etc. So I found: ∂F/∂T = 1/2.aM^2 ∂F/∂M = TaM - TcaM + bM^3 And using a chain rule: dF/dM = ∂F/∂T.dT/dM + ∂F/∂M.dM/dM But not sure how to find dT/dM
The statement of the problem says find ##\frac{dF}{dM}##. Isn't the expression just a polynomial in ##M##? Hold everything else constant and differentiate it.

 Quote by LCKurtz Is that supposed to be$$F=(\frac 1 2)a(T-T_c)M^2 +(\frac 1 4)bM^4$$
Yes that's the correct equation.

Isn't T also a variable or have I just forgotten how to do basic differentiation?

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## Multi-variable differential question

 Quote by steejk 1. The problem statement, all variables and given/known data F = 1/2.a(T-Tc)M^2 + 1/4.bM^4 I need to find dF/dM a,Tc,b are positive constants
 Quote by steejk Yes that's the correct equation. Isn't T also a variable or have I just forgotten how to do basic differentiation?
You said it was a constant above. And if it weren't, you would still calculate the partial derivative ##\frac{\partial F}{\partial M}## the same way.

 Quote by LCKurtz You said it was a constant above. And if it weren't, you would still calculate the partial derivative ##\frac{\partial F}{\partial M}## the same way.
Sorry, Tc is a constant but not T.

I can see the partial derivative would hold everything else constant, but I need to find the total derivative dF/dM.
 Recognitions: Gold Member Homework Help I guess I misunderstood what you wanted. If T depends on M, then the chain rule in your original post would be correct. But without more information, I don't see you you could calculate ##\frac{dT}{dM}##.

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