One of the conditions on [itex]\nabla[/itex] mentioned in the definition of a connection is that
[tex]\nabla_X(fY)=(Xf)Y+f\nabla_XY[/tex]
Note that Xf and fX are not the same. Xf is the function [itex]p\mapsto X_pf[/itex] and fX is the vector field [itex]p\mapsto(p,f(p)X_p)[/itex]. The effect of [itex]\nabla_X[/itex] on a function is defined as
[tex]\nabla_Xf=Xf[/tex]
Use this definition with [itex]X=\partial_\mu[/itex], and you get
[tex]\nabla_{\partial_\mu}f=\partial_\mu f[/tex]
so
[tex]t(f)=t^\mu\partial_\mu f=t^\mu\nabla_{\partial_\mu}f[/tex]
Note also that the definition allows us to rewrite the first equation as
[tex]\nabla_X(fY)=(\nabla_Xf)Y+f\nabla_XY[/tex]
The other two requirements in the definition of a connection are
[tex]\nabla_X(Y+Z)=\nabla_XY+\nabla_XZ[/tex]
[tex]\nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ[/tex]
