while discussing 1-forms, it is given
Our goal is to generalize the concept of the gradient of a function to functions on arbitrary manifolds. What we will do is to make up, for each smooth function $f$ on $M$, an object called $d f$ that is supposed to be like the usual gradient $\nabla f$ defined on $\mathbb{R}^n$. Remember that the directional derivative of a function $f$ in the on $\mathbb{R}^n$ in the direction $v$ is just the dot product of $\nabla f$ with $v$ : $$ \nabla f \cdot v=v f \qquad ............... (1) $$ In other words, the gradient of $f$ is a thing that keeps track of the directional derivatives of $f$ in all directions.
I didn't get how the equation (1) is arrived at. I assume $v$ is a vector and $vf$ is product of $v$ with the scalar $f$. Please pardon me if this is trivial.