I don't know if derivation is the right expression but we have the following to solve:
Let where p is a point of a smooth manifold M. Furthermore assume that f(p)=g(p)=0. Now show that for all v(fg)=0.
According to the lecture we defined to be the function germ in point p. was defined as the tangential vector in p that is a function . A tangential vector v fulfills:
(1.) if f stationary
(2.) For scalars and f,g function germs we have
I try to go by contradiction. f,g are .
v(fg)=v(f)*g(p)+f(p)*v(g)=v(f)*0+0*v(g)=0 by the derivative rule.
Is this the solution already?