In Proposition 13.11 of Lee's Smooth Manifolds book he asserts that for smooth $k$-form $\sigma$ and vector fields $X, Y_1,...,Y_k$ we have $\mathcal{L}_X(\sigma(Y_1,...Y_k)) = (\mathcal{L}_X\sigma)(Y_1,...,Y_k) + \sigma(\mathcal{L}_XY_1,...,Y_k) + ... + \sigma(Y_1,...,\mathcal{L}_XY_k) $. He provides no proof, and I am not finding it anywhere near so straightforward as he asserts-- we can use $\mathcal{L}_X(\sigma(Y_1,...Y_k)) = X(\sigma(Y_1,...,Y_k)) $ but it is not at all clear how to proceed from here. Does anyone have a proof of this result?
(Note: $\mathcal{L}_X$ is the Lie Derivative, here defined as $\mathcal{L}_X(\sigma) = \lim_{t \to 0}\frac{(\theta_t)^*\sigma_{\theta_t(p)} - \sigma_p}{t}$ where $\theta$ is the flow of $X$)
There is nothing about Lie derivatives in Proposition 13.11, in either the first or the second edition of my Introduction to Smooth Manifolds. You must be looking at one of the pirated draft versions of the first edition, which somebody posted illegally on the internet. Those are full of mistakes and come with no guarantees.