How do I show that $\partial/\partial x^i |_p$ satisfies the derivation property?

Question

I am studying Tu's An Introduction to Manifolds and he says that "it is easily checked that $\partial/\partial x^i |_p$ satisfies the derivation property and so is a tangent vector at $p$" where $$\frac{\partial}{\partial x^i}|_p f = \frac{\partial}{\partial r^i }|_{\phi(p)} (f\circ \phi^{-1}) \in \mathbb{R}.$$ I tried applying the definition but I don't know how to use $$\frac{\partial}{\partial r^i }|_{\phi(p)} (fg\circ \phi^{-1})$$ because of the $\phi^{-1}$ term to show that $$\frac{\partial}{\partial x^i}|_p(fg) = \left(\frac{\partial}{\partial x^i}|_pf\right)g(p) + \left(\frac{\partial}{\partial x^i}|_pg\right)f(p).$$ Can someone help me see what I'm missing?