Understanding the increment in Directional Derivative

I am trying to make sense of the Directional Derivate formula I have been given at class.

Having a differentiable function and a unit vector, we define the Directional Derivative as:

$$ \frac{f(x+h\cdot u_1,y+h\cdot u_2)}{h} $$

I am having trouble understanding the increment h in here, since coming from the Partial Derivatives, in this case, respect x

$$ \frac{f(x+h ,y)}{h} $$

the h is the increment in the x axis, or better said, the "distance" between x and x'. I don't think the distance in the Partial Derivatives is h.

What am I missing?

Thanks

Solution 1:

In your difference quotient $h$ is the change along the direction of the unit vector $(u_1, u_2)$.

The partial derivative is the special case when the unit vector is $(1,0)$.