The Second Order Derivative f_xx (x, y)

We have already established that f _x (x, y) represents the slope in the x direction. If the derivative of f _x (x, y) with respect to x is taken, the result is:

Geometrically, this is the rate at which the slope in the x direction is changing as we move in the x direction.

To visualize this geometrically, we can start by observing the graph of f (x, y) = x² + y² shown in the graph below.

By taking the partial derivate with respect to x and evaluating it at the appropriate point, we can conclude that
(0, 0, 0) the slope in the x direction is 0.

By taking the partial derivate with respect to x and evaluating it at the appropriate point, we can conclude that at
(2, 0, 4), the slope in the x direction is 4.

Correspondingly, the change in f_x (x, y) is 4 as we move 2 units in the x direction.

Hence, we can conclude that