Covariant derivative on the dual bundle The covariant derivative on the dual bundle is defined as follows: 

$\nabla^{*}: \Gamma(TM) \times \Gamma(E^*) \ni (X, t) \mapsto \nabla_X^{*} t \in \Gamma(E^*)$

, where for any section $s \in \Gamma(E)$, 

$(\nabla_X^{*} t)(s) = L_X(t(s)) -  t(\nabla_X s)$.

Remark: $\Gamma(E^*)$ is the set of sections of the dual bundle and $\Gamma(TM)$ is the set of vector fields. 
I would like to check whether this is indeed a covariant derivative. I have already proved that it satisfies function-linearity, but now I have difficulties to show that it satisfies Leibniz rule, i.e. $(\nabla_X^* ft) = (L_X f)t + f\nabla^*_X t$. 
I need to verify that 
 $(\nabla_X^* ft)(s) = L_X(ft(s)) - ft(\nabla_X s) = \ldots = (L_X f)t(s) + f(\nabla^*_X t)(s). $ Can someone help me to find the intermediate steps ? 
Thanks. 

As written in the post,
$$\nabla^*_Xft(s)=X(ft(s))-ft\nabla_Xs.$$
Now, using the Leibniz rule for the first summand on the right, 
$$=X(f)t(s)+fX(t(s))-ft\nabla_Xs=X(f)t(s)+f\nabla^*_Xt(s).$$
