I'm reading First Steps in Differential Geometry and the author gives the following formula without proof: $$\mathcal{L}_X(dx_i)=\displaystyle\sum_{j=1}^n\frac{\partial X^i}{\partial x_j}dx_j$$ where $X$ is a vector filed and $dx_i$ is a $1$-form. I need to prove it by the definition that $$\mathcal{L}_XT=\frac{d}{dt}\bigg|_{t=0}(\phi_t^*T)$$ where $T$ is a tensor field and $\phi_t$ is a flow induced by a vector space $X$.
I tried to show it through computing $\mathcal{L}_X(dx_i)(Y)(x)$ where $Y$ is another vector field but failed, may be it is totally wrong.(Actually I'm not quite understand the meaning of $\phi_t$ and $\phi_t^*$, which makes it difficult for me to calculate.)
By definition, $d$ and $\mathcal L_X$ commute, since $d$ commutes with pullback of maps (in this case the map is $\phi_t$). But then $$ L_X (dx^i) = d(L_X x^i) = d(X(x^i)) = d\left(\sum_k X_k \frac{\partial x^i}{\partial x^k}\right) = d(X_i) = \sum_k \frac{\partial X_i}{\partial x^k}dx^k. $$