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# chain rule, two variables watch

1. Let x and y denote Cartesian and polar coordinates. F(r,th) = f(x,y)

How would I obtain the second differential of f wrt x?

I've differentiated once with the chain rule to get

(dF/dr)(dr/dx) + (dF/dth)(dth/dx)

then I dont know how do differentiate again

I've used product rule but I get terms like

d/dx(dF/dr) which I'm not sure what to do with

many thanks
2. Well, d/dx = (dr/dx)d/dr + (dt/dx)d/dt, no?
3. If u = f(x, y) then du/dr = (df/dx)(dx/dr) + (df/dy)(dy/dr) = cos (th)(df/dx) + sin (th)(df/dy)

get du/dth and solve for df/dx and df/dy

then you will have the operator d/dx so you can do d/dx(df/dx)

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Updated: November 15, 2008
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