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Simple Differentiation?


In Mathematica 5.0, if one types in:

f[x_]+g[x_]^:=1   (1)

and then types

dx(f[x]+g[x])

we get 0.

However, if we type dx(f[x]) + dx(g[x]) after typing (1), we get

f'[x] + g'[x].

Mathematica doesn't recognize that f'[x]+g'[x]=
dx(f[x]+g[x]), which equals 0. How does one get Mathematica to do this?

I'm interested in more complicated examples, which involve more complex
differential relations, but I should be able to do things in those
situations given a nice, natural way of handling this toy case. Though this
seems like a pretty simple thing, I wasn't able to find any thing in the
Mathematica Book or elsewhere, which would help with this type of things.

Thanks,
Sunil



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