= Solution
Define the <Lie derivative of a differential form> by $\mathcal L_X\alpha=(d/dt)_{t=0}\phi_t^*\alpha$, using the <local flow>. The <pullback of a differential form> respects the <wedge product of differential forms> and commutes with the <exterior derivative>. Therefore $\mathcal L_X$ is a <derivation of an algebra> of degree zero and commutes with $d$.
Let $D_X=\iota_Xd+d\iota_X$, where $\iota_X$ is the <interior product of a differential form>. The graded product rules for $d$ and $\iota_X$ show that their anticommutator $D_X$ is also a degree-zero <derivation of an algebra>: the two mixed terms cancel. Since $d^2=0$, it commutes with $d$. On a function $f$, $\iota_Xf=0$ and $D_Xf=\iota_Xdf=Xf=\mathcal L_Xf$. Hence $D_X(df)=d(Xf)=\mathcal L_X(df)$.
In a <manifold chart>, every <differential form> is a sum of functions times products $dx^{i_1}\wedge\cdots\wedge dx^{i_p}$. The two derivations agree on its generators $f$ and $dx^i$, and therefore agree on every such form. This proves <Cartan's magic formula>, including the case $p=0$:
$$
\boxed{\mathcal L_X\alpha=\iota_Xd\alpha+d(\iota_X\alpha).}
$$
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