Let $M$ be a manifold. The differential of $f \in C^\infty(M)$ is defined as the one-form $df$ such that $(df)(v) = v(f)$ for every tangent vector $v$ to $M$.
It is then stated that if $f \in C^\infty(M)$ and $h \in C^\infty(\mathbb{R})$, then $d(h(f)) = h^\prime(f)\,df$. I don't know what this notation means. Will someone please write it down more precisely (see comments)?
Edit: Using the regular definition of the differential, we have $$d(h \circ f)_m(v) = dh_{f(m)} \circ df_m(v) = dh_{f(m)}\left(v(f)\left.\frac{d}{dt}\right|_{f(m)}\right) = v(f)h^\prime(f(m)),$$ and then writing $v(f) = df_m(v)$ gives the result. However, I'm not sure why it is appropriate to use two different definitions of the differential in the same equation.