Khan Academy's article on the directional derivative is very helpful to me for my understanding. However, there is a section of it that I'm confused about, namely this part:
It is not apparent to me that Step 3 changes the output of the function by $hv_1\left(\frac{\partial f}{\partial x}(x_0,y_0)\right)$, and their article hyperlinked that states to explain it doesn't for me. Does someone know where they're coming from here?
Note that the expression$$hv_1\left(\frac{\partial f}{\partial x}\right)(x_0,y_0)$$ is a product of two quantities.
The first part is $hv_1$ which is simply the change in $x$.
The second part $\frac{\partial f}{\partial x} (x_0,y_0)$ which is the rate of change of $f$ with respect to $x$.
When we multiply the $x$-increment by the rate of change of the function with respect to $x$, we get the total change of $f$ due to this $x$-increment.