Partial differential derivation

[Taylor_1989]

Homework Statement

Hi guys, I am having a problem, knowing where to start with this question. Before I spend trying derive the partial derivative chain rule from first principles I would just like to know if this is what this questions is asking. If it is not asking that, how do I go about solving it.

Homework Equations

The Attempt at a Solution

I have not shown solution beacuse I am unware where to start

 

[Taylor_1989]

I have mange to come up with one solution, which: If I think that y,x are all functions of t. Then I could say: dx=dx/dt *dt and dy=dy/dt *dt sub into the total differential and get $\partial{df}{dx}\frac{dx}{dt}*dt+\partial{df}{dy}\frac{dy}{dt}*dt$ I am just unsure of this method, I feel like I am cheating, any advice?

 

[Mark44]

I have mange to come up with one solution, which: If I think that y,x are all functions of t. Then I could say: dx=dx/dt *dt and dy=dy/dt *dt sub into the total differential and get $\partial{df}{dx}\frac{dx}{dt}*dt+\partial{df}{dy}\frac{dy}{dt}*dt$ I am just unsure of this method, I feel like I am cheating, any advice?

This -- $\partial{df}{dx}\frac{dx}{dt}*dt+\partial{df}{dy}\frac{dy}{dt}*dt$ -- doesn't make any sense. The expression $df$ is defined (it's the differential of f), but this one $\partial f$ doesn't mean anything.

You have f(x, y) where x is a function of t and y is another function of t. This means that f is ultimately a function of t, albeit one with two parameters.

 

[Taylor_1989]

Sorry for the bad latex, I did eventually solve the problem. I was over thinking the problem and thought I had to derive partial chain rule from first principles, which I on my way to doing, by deriving the non partial chain rule. As a question, am I on the right lines if I derive chain rule from first principles then, then apply the same method to partial, would I be able to derive the above formula?

 

[Mark44]

Sorry for the bad latex, I did eventually solve the problem. I was over thinking the problem and thought I had to derive partial chain rule from first principles, which I on my way to doing, by deriving the non partial chain rule. As a question, am I on the right lines if I derive chain rule from first principles then, then apply the same method to partial, would I be able to derive the above formula?

Which formula above do you mean?