Directional derivative [closed]

1. What does a directional derivative tell you?
2. Does the directional derivative always exist?
3. What happens if the directional derivative is 0?
4. How do you prove directional derivative does not exist?

What does a directional derivative tell you?

Directional derivatives tell you how a multivariable function changes as you move along some vector in its input space.

Does the directional derivative always exist?

Therefore directional derivatives in all directions exist. The vector (fx(X0),fy(X0),fz(X0)) is called gradient of f at X0 and is denoted by ∇f(X0).

What happens if the directional derivative is 0?

The directional derivative is a number that measures increase or decrease if you consider points in the direction given by →v. Therefore if ∇f(x,y)⋅→v=0 then nothing happens. The function does not increase (nor decrease) when you consider points in the direction of →v.

How do you prove directional derivative does not exist?

Take a direction uθ=(cosθ,sinθ). Hence the directional derivative (limit of above for h→0) cannot exists except for θ∈kπ2 ; k∈Z, I.e. in the x or y direction.