Let $f: M \to K$ be a local diffeomorphism between manifolds. We define the pullback of a vector field $\xi$, denoted as $f^{*}\xi : M \to TM$, where $f^{*}\xi = (T_{x}f)^{-1}\xi(f(x)) \in T_{x}M.$
The professor had in their notes that for $h \in C^{\infty}(K, \mathbb{R})$ we have that $$\begin{equation} f^{*}(h\xi) = (h \circ f)f^{*}\xi \quad \forall \xi. \end{equation}$$
By plugging it in the equation - $$\begin{equation} (T_{x}f)^{-1}h\xi(f(x)) = (h \circ f)(T_{x} f)^{-1}\xi(f(x)). \end{equation}$$
Maybe I'm not seeing it, but I'm not sure how to proceed at the moment. Is there a tip I should be aware of? Thanks in advance!
Note that by pointwise multiplication we have
$$(h\xi) (f(x)) = (h\circ f)(x)\, \xi(f(x))$$
So by linearity of $(T_xf)^{-1}=T_{f(x)}f^{-1}$ the claim follows, i.e.
$$(T_x f)^{-1} (h\xi)(f(x))= (T_x f)^{-1}\left[(h\circ f)(x)\, \xi(f(x))\right] = (h\circ f)(x)\,(T_x f)^{-1}\xi(f(x))= ((h\circ f)\,f^*\xi) (x)$$