What Is the Directional Derivative of a Function at a Point?

[lonewolf219]

Homework Statement

f(x.y)=4x^2-y^2

Homework Equations

Ʃ partial derivative components(?)

The Attempt at a Solution

The solution when θ=pi and f(1,-1) is -8.

Does this mean that one of the coordinates of this function is (1,-1,-8)?
What exactly is the directional derivative, and what does the solution represent?

 

[LCKurtz]

Homework Statement

f(x.y)=4x^2-y^2

Homework Equations

Ʃ partial derivative components(?)

The Attempt at a Solution

The solution when θ=pi and f(1,-1) is -8.

Does this mean that one of the coordinates of this function is (1,-1,-8)?
What exactly is the directional derivative, and what does the solution represent?

You haven't stated the problem for which you are giving the solution. I'm guessing it was "Find the directional derivative of f(x,y) at the point (1,-1) in the direction of $\theta=\pi$. To help you visualize what you are calculating, think of a flat metal plate and suppose $f(x,y)=4x^2-y^2$ as the temperature at each point in the plate. If you were at (1,-1) the temperature there would be f(1,-1) = 3. Depending on what direction you move from that point, it may get warmer or colder. The directional derivative in some direction at that point is the rate of change of temperature in that direction. So according to your calculations above, if you move in the $\pi$ direction from there it is cooling off at 8 degrees / unit length.