I'm trying to learn a bit about differential forms to supplement my study in analysis, but I'm having a hard time with some of the basic manipulations.
Anyway, suppose $\Omega$ is an open set in $\mathbb{C}$, and $\varphi:\Omega\to\mathbb{C}$ a smooth map. For a function $f$, I have the definition $\varphi^*f=f\circ\phi$, (when this makes sense for $f$ of course).
I also have the definitions $ \varphi^*\,dx=\frac{\partial\varphi_1}{\partial x}\,dx+\frac{\partial\varphi_1}{\partial y}\,dy, \qquad \varphi^*dy=\frac{\partial\varphi_2}{\partial x}\,dx+\frac{\partial\varphi_2}{\partial y}\,dy, $ where $\varphi_1$ is the $x$ component of $\varphi$ and $\varphi_2$ is the $y$ component. For a $1$-form $h=f\,dx+g\,dy$, $ \varphi^*h=(\varphi^*f)\varphi^*\,dx+(\varphi^*g)\varphi^*\,dy. $
What I don't get is why how $d(\varphi^* f)=\varphi^*df$.
I thought $d(\varphi^*f)=d(f\circ \phi)=f\,d\varphi+\varphi \,df$. I think my big problem here is I don't understand how to apply $d$ to a composition of functions. I also think $ \varphi^*(df)=\varphi^*\left(\frac{\partial f}{\partial x}\,dx+\frac{\partial f}{\partial y}\,dy\right) $ but I don't know how to take this further. They don't look equal to me. How does equality follow here? Thank you.