# Another Directional Derivative

• Sep 13th 2011, 11:19 AM
dbakeg00
Another Directional Derivative
I'm new at working with directional derivatives. I'm stuck on this problem. Would someone mind helping me with the "Point" and "Direction" parts of the problem. I'm just looking for a place to start, not for someone to do the problem for me. Thanks.

In each of the following, find the directional derivative of f at P in the given direction.

1. f(x,y)=xy(1-x^2-y^2), P=any point on the unit circle, Direction: in the direction of OP.

I found that the gradient of f is: <y(1-x^2-y^2)-2yx^2, x(1-x^2-y^2)-2xy^2>, so I'm stuck on finding P and the direction.
• Sep 13th 2011, 11:42 AM
Plato
Re: Another Directional Derivative
Quote:

Originally Posted by dbakeg00
In each of the following, find the directional derivative of f at P in the given direction.
1. f(x,y)=xy(1-x^2-y^2), P=any point on the unit circle, Direction: in the direction of OP.
I found that the gradient of f is: <y(1-x^2-y^2)-2yx^2, x(1-x^2-y^2)-2xy^2>, so I'm stuck on finding P and the direction.

If $P: (a,b)$ then $\overrightarrow {OP}=$ and you know that $a^2+b^2=1$.
• Sep 13th 2011, 12:42 PM
dbakeg00
Re: Another Directional Derivative
Ok, I did the problem, would you mind checking for me? Thanks

So I changed [the format] of f (x,y) = (xy-x^3y-xy^3)
and P=(a,b) & OP=<a,b>

Now I have the gradient of f=<y-3x^2y-y^3, x-x^3-3xy^2>

Duf(x,y)=(a)(y-3x^2y-y^3)+(b)(x-x^3-3xy^2)

Duf(a,b)=(a)(b-3a^2b-b^3)+(b)(a-a^3-3ab^2)

Duf(a,b)= (ab-3a^3b-ab^3) + (ab-a^3b-3ab^3)

Duf(a,b)= 2ab-4a^3b-4ab^3

Duf(a,b)= ab-2a^3b-2ab^3

Duf(a,b)= ab(1-2a^2-2b^2)