Let $C^1_0\mathbb{R}$ denote the algebra of germs of $C^1$-functions in a neighbourhood of $0$ in $\mathbb{R}$. Furthermore let $I$ be the subalgebra of germs in $C^1_0\mathbb{R}$ that vanish at $0$ and $I^2$ be the subspace of $I$ consisting of linear combinations of products of elements of $I$. If $h \in C^1$ in a neighbourhood of $p$ let $[h]$ denote its germ.
Show that if $[h] \in I^2$, then $h'(0) = 0$ for any representative of $[h]$.
So $h \in C^1$ defined on a neighbourhood of $0$ and $h(0) = 0$.I tried to use the definition, i.e. $$h'(0) := \lim_{\varepsilon \to 0} \frac{h(\varepsilon) - h(0)}{\varepsilon} = \lim_{\varepsilon \to 0} \frac{h(\varepsilon)}{\varepsilon}$$ which does not really help. Has anyone a hint for me?
Hint: You don't need the definition of the derivative, just the formula for the derivative of a product. Think about what it means that $[h] \in I^2$, not just $I$.