I'm currently working through a paper that uses quite a bit of multivariable calculus, which I am admittedly quite rusty at. At one point a line integral is taken over a straight path between two points, and then the directional derivative is applied. I simplified the scenario for the purposes of this question:
Let $AB$ be the straight line segment connecting two points $A$ and $B$ in the plane, and $\bar\nabla$ be the directional derivative in the direction of $AB$. Define: $$I(A) = \int_{AB} f\big(\vec{t}\big)d\vec{t}$$
Claim: $$\bar\nabla I = -f(A)$$
My question: where is the negative sign coming from? In the one dimensional case this is a consequence of the fundamental theorem of calculus along with flipping the bounds of integration. I'm assuming a similar idea applies here, but I am having trouble justifying this. Any help would be much appreciated!