Directional Derivative in direction of max change

I need help! I am suppose to find the directional derivative of function xy + z^2 at point (2,3,4)

I know that this means in the direction of the gradient of the function.

So., using the partial derivatives of xy + z^2 which are y , x, 2z I have y ivectors + x j vectors + 2z k vectors.

When I come up with my vector in the greatest direction and then the unit vector for this and multiple it times the 3 i + 2 j + 8 k I get a value of 1!!!! I must be doing something wrong but I do not know what!!!! Frostking in a blizzzzzard of confusion.

Directionn derivative in direction of max change

The direction is that of the maximum change so I thought this was in the direction of the gradient of the function and when I used this I got 1 for an answer. Does this clarify? Thanks for helping!!!!