On the properties of the covariant derivative

Question

In Do Carmo's textbook "Riemannian Geometry", he defines the covariant derivative on a manifold $M$ as a mapping that satisfies the following properties:
$1) \nabla_{fX+gY}Z=f\nabla_XZ+g\nabla_YZ$
$2) \nabla_X(Y+Z)=\nabla_XY+\nabla_XZ$
$3) \nabla_X(fY)=X(f)Y+f\nabla_XY$
for $X, Y, Z\in \mathfrak{X}(M)$ and $f, g$ are $C^{\infty}$ on $M$.

My question is the following:
What is wrong with writing $(2)$ and $(3)$ as one property, namely:
$\nabla_X(fY+Z)=X(f)Y+f\nabla_XY+\nabla_XZ$ ?

Thank you in advance.

Answer 1

You can do that. It is just a matter of convenience. Indeed, if you choose $f=1$ your property reduces to (2) and if you choose $Z=0$ your property reduces to (3). You can even incorporate all the three into one property, but that would be just a mess.