There's a cute proof of this in Spivak's "Calculus on Manifolds". If you want to try it yourself, try defining the function fd(x) = f(x + h) - f(x) - df(h) and similarly for g, pretty much the numerator in the definition of the derivative, then evaluating the limit in the definition of the derivative for [itex]g\circ f[/itex], where you let the linear function in the limit be dg(f(x))df(x).
yea i thought of that and thought the same thing; that there's gotta be a more rigorous proof.
more rigorous than very rigorous
maybe you mean more detailed
that I can provide
with E (or a subset) open in F
F (or a subset) open in G
and f differentiable at x
g differentiable at f(x)
(g◦f)' exist with
or in more full notation
notes on mappings
where L(E,F) is a space of linear mappings from E to F
so all is as it should be
thus derivatives are linear mappings
Δf=df+o(dx) (f differentiable)
so lim rf=0
lim rf=lim rg=0 (f,g differentiable)
we now need only
which is clear from
ie bounded near x
where |f'(x)| is the norm induced on linear maps by the norm on vectors
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