Question

I am really confused about why when given a function f(x,y) where x = g(t) and y = h(t) then dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). I get that this is an application of the chain rule but could someone please describe the proof if possible?

Answer 1

it might help jog my moemory if you can work out the proof to where you get stuck at

Answer 2

if not then this might be useful

http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx

Answer 3

that helps a little but I am still confused about how they went from the first form of the chain rule to the second form.

Answer 4

As in how do partial derivatives relate at all?

Answer 5

\[z(t) = g(t)~h(t)\]
\[\frac {dz}{dt} = \frac{dg}{dt}h(t)+g(t)\frac {dh}{dt}\]
yeah, still trying to recall some of this :)
\[\frac{dz}{dt}=\frac{dz}{dg}\frac{dg}{dt}y+x\frac{dz}{dh}\frac{dh}{dt}\]

Answer 6

that last setup is off ...

Answer 7

but how do you know z(t) = g(t)*h(t) when it could be z(t) = g(t) + h(t) or z(t) = g(t)/h(t) or an infinite number of other things.

Answer 8

ill prolly have to back step into this .. but lets assume we already know the proofing to a partial derivative setup

\[z=f(x(t),y(t))\]\[z_x=f_x(x(t),y(t))~x'(t)~;~z_y=f_y(x(t),y(t))~y'(t)\] OR
\[\frac{\delta z}{\delta x}=\frac{\delta f}{\delta x}\frac{dx}{dt}\]\[\frac{\delta z}{\delta y}=\frac{\delta f}{\delta y}\frac{dy}{dt}\]

Answer 9

http://www.youtube.com/watch?v=JSs_dqq2uWo

you might get some better insight from Herbert Gross. I got no audio at the moment to refresh my memory, but i find this guy to be awesome.

Answer 10

thanks for all your help @amistre64. i will watch the video. I am still kind of confused but I definitely appreciate all of the time you have devoted to my question.

Answer 11

youre welcome, im gonna rewatch Dr Gross when i have a better setup. seems like i need the refresher :)