The directional derivative of a function $f$ is taken "along" some vector $\vec v$ in $f$'s input space, and is denoted $f_{\vec v}'$, according to Khan Academy
Multivariable functions also have partial derivatives, which are denoted $f_x$ for example.
But in my textbook, I"m seeing this symbol: $f_{\vec u}$. It lacks the $'$ symbol that is present in a directional derivative, but it also is with respect to a vector unlike a partial derivative.
What is this symbol representing? For context, this appeared in the derivation for the Fundamental Theorem of Calculus for Line Integrals. Here is the part where the symbol appears: 
What does $f_{\vec u}$ refer to above? I know it refers to the "rate of change of $f$", but what type of operation is being specified with this notation?
We have that $ f_{\hat u}$ represents the directional derivative in the direction of vector $\vec {\Delta r_i}$ that is $\nabla f\cdot \hat u$ where $\hat u=\frac{\vec {\Delta r_i}}{|\vec {\Delta r_i}|}$.