I was reading through the Wikipedia page (https://en.wikipedia.org/wiki/Lie_derivative) for a Lie Derivative, and it states the following property in the bottom of the section "The Lie Derivative of a differential form"
$L_{fX}w = fL_X w + df \wedge i_X w$
I haven't seen this stated before, does anybody know of a textbook that proves this? It seems like it follows from Cartans identity but I was struggling to prove it to myself.
Here's an approach which doesn't use the local coordinates or Cartan's formula, at least directly. First assume $\omega$ is a 1-form, then the formula you've written is a formula of 1-forms, so it is true iff it is true upon testing on arbitrary vectors fields $Y$. Testing on $Y$ is allows us convert from a Lie derivative of a 1-form to a Lie derivative of a vector field which is much friendlier to deal with.
Since Lie derivative is a derivation, it satisfies the following product rule $$X(\omega(Y))=(\mathcal{L}_X \omega)(Y)+\omega(\mathcal{L}_X Y).$$ Therefore, we compute \begin{align*} (\mathcal{L}_{fX}\omega)(Y)&=(fX)(\omega(Y))-\omega(\mathcal{L}_{fX}Y)\\ &=f(X(\omega(Y)))-\omega(f\mathcal{L}_XY-(Yf)X) \\ &=f(X(\omega(Y))-\omega(\mathcal{L}_XY))+(Yf)\omega(X)\\ &=f(\mathcal{L}_X \omega)(Y)+df(Y)\wedge\iota_X \omega, \end{align*} where the second equality is from the computation \begin{align*} \mathcal{L}_{fX}Y&=(fX)Y-Y(fX)\\ &=f(XY)-(Yf)X-f(YX)\\ &=f[X,Y]-(Yf)X\\ &=f(\mathcal{L}_XY)-(Yf)X. \end{align*}
Now the general case, as usual with these proofs, is nothing more than bookkeeping. If $\omega$ is an $n$-form, then we must test on $n$ vector fields $Y_1,...,Y_n$. We compute \begin{align*} (\mathcal{L}_{fX}\omega)(Y_1,...,Y_n)&=(fX)(\omega(Y_1,...,Y_n))-\sum_{i=1}^n\omega(Y_1,...\mathcal{L}_{fX}Y_i,...,Y_n)\\ &=f(X(\omega(Y_1,...,Y_n)))-\sum_{i=1}^n\omega(Y_1,...,f\mathcal{L}_XY_i-(Y_if)X,...,Y_n)\\ &=f(\mathcal{L}_X\omega)(Y_1,...,Y_n)+\sum_{i=1}^n\omega(Y_1,...,(Y_if)X,...,Y_n)\\ &=f(\mathcal{L}_X\omega)(Y_1,...,Y_n)+\sum_{i=1}^n df(Y_i)\wedge\omega(Y_1,...,X,...,Y_n)\\ &=f(\mathcal{L}_X\omega)(Y_1,...,Y_n)+(df\wedge \iota_X\omega)(Y_1,...,Y_n). \end{align*}