I'm still trying to grasp fully the concept of directional derivatives and how they can be used to approximate functions, and so I have an example of a problem I made up that I tried to evaluate. Let's say I have the function,
$$f(x,y) = x^2+y^2$$
I want to evaluate the directional derivative at the point P = $(2,2,8)$ with the vector $\vec{v} = (1,2)$.
So I first found the gradient $\nabla f = (2x,2y)$, and plugging in the original points I got the gradient at point P to be $(4,4)$. I then found $\nabla f \cdot P$ and got $4 * 1 + 4 * 2 = 12$. So the directional derivative would be $12$. My understanding is that this is the best approximation around the point, and says that moving by $h*\vec{v}$ will cause a change of $12 * h$. So when $h=1$, you would be moving to the point (2,3) and the approximation would be $f \approx 8 + 12 = 20$, but the actual value of the function is $f = 25$. That's a very large error, but when $h$ gets smaller and smaller, this error will decrease monotonically.
I want to clarify if my logic, calculation, and reasoning is correct before I move on from this topic. I'd be worried if I went into the next topic with a fundamental misunderstanding.