Does indicate the derivative of f in the direction of unit vector u? Yes, you are right. The derivativein a specific directionmay exist even if the function is not "differentiable".

You can prove it directly from the definition:

The definition of the derivative, in the direction of vector u, is

It follows that .

Multiply numerator and denominator by c to get

Now let, k= ch. Since c is a constant, k goes to 0 as h goes to 0 and we can write that

.