# Is the partial derivative of a function with respect to a vector different than the directional derivative?

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 $$\hat u=\frac{\vec {\Delta r_i}}{|\vec {\Delta r_i}|}$$.
• Thank you for the help! Sorry for being slow, a couple questions: 1. Is there a reason there's a vector arrow above $f$? Is $f$ not a scalar valued function? (assuming the arrow implies vector valued). 2. So are you saying that $\nabla f \cdot \hat u = \vec {\Delta r_i}$? And $f_{\hat u}$ denotes the directional derivative? Does the lack of a $'$ have any significance? – James Ronald Nov 9 '19 at 1:27
• @JamesRonald opsss sorry the arrow on $f$ is just a typo! I fix that. – user Nov 9 '19 at 1:50
• @JamesRonald In teh context here presented $f_{\hat u}=\nabla f \cdot \hat u$ represents the directional derivative in direction of the vector $\Delta \vec r_i$ which has unitary direction vector $\hat u$. – user Nov 9 '19 at 1:53