I am having trouble understanding this.
Start with 1 variable:
If I have a function say, f(x) x^4-3x^3+2 that has a derivative 4x^3-9x^2 the gradient (here just the derivative) at a starting value of say x=-6 is -1188.
(1) Does this tell me that the largest rate of change (increase) occurs in the direction of -1188? Is this X or Y?
(2) To get steepest descent, I move in the opposite direction, towards 1188. What exactly does it mean to move in the direction of 1188? It does not seem to mean "go to point x=1188". We take x=6 - (-1188). This is the starting point to plug into the gradient. I dont think I understand.
If you are working in 2+ variables, same questions:
(1) what does it mean to "move in the direction of a vector"?
(2) why take where we were (x1,y1) when working in 2 variables and subtract from that the gradient evaluated at (x1,y1)?
I would be appreciative if someone can explain this better than my book!