I'll just get straight into the question then say what I'm confused about at the bottom...
x = rcos(t) and y = rsin(t), find dr/dx and dr/dy (partial derivatives)
The way to solve it in my notes goes:
dx = (dx/dr)dx + (dx/dt)dt = cos(t)dr - rsin(t)dt (sorry, don't know how to do partial d's)
cos(t)dx = cos^2(t)dr -rsin(t)cos(t)dt
dy = sin(t)dr + rcos(t)dt
sin(t)dy = sin^2(t)dr + rsin(t)cos(t)dt
so: cos(t)dx + sin(t)dy = dr
= (dr/dx)dx + (dr/dy)dy
-> (dr/dx) = cos(t) and (dr/dy) = sin(t)
I get all that, but if we go back to x = rcos(t) and y = rsin(t), couldn't we just rearrange these to make r the subject, the take the partial derivatives of that? But this gets (dr/dx) = 1/cos(t) and (dr/dy) = 1/sin(t), so it obviously can't be right, but where is it incorrect? I'm probably missing something simple...