I've seen the formula for differentials alot, namely
dy=f′(x)dx
but what I think when I see this is that someone is manipulating the "formula"
f′(x)=dydx
When I think of "differential equation", I think of
f(x,y,dydx,d2ydx2,⋯,dnydxn)=0
not
f(x,y,dy,dx,⋯)=0
I've heard that Δy and Δx can be approximated by dy and dx (or maybe its the other way around?), but that doesn't make much sense to me. If you replace dy and dx by Δy and Δx, you sort of have Euler's method, but this still doesn't clear much up for me. So,
What exactly is a differential, and why is it useful?
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