Alright, I'm trying to fine the gradient of the function and the maximum value of the directional derivative.
$z = \frac{y}{(x^2 + y^2 )}\text{ at }(1,1)$
So far, I've gotten the following values for partial derivates, but I can't guarantee they are correct:
$F_x = \frac{-2xy}{(x^2 + y^2 )^2}$ $F_y = \frac{-2y^2}{( x^2 + y^2 )^2} + \frac1{x^2 + y^2}$
GradientF(x,y) = \frac{-2xy}{( x^2 + y^2 )^2 + (-2y^2)/( x^2 + y^2 )^2} + 1/( x^2 + y^2 )^2}$$ GradientF(1,1) = -2/4i + (-2/4)j + (1/2)k = <-2/4, -2/4, 1/2> ||F(1,1)|| = <-2/4, -2/4, 1/2> (1/sqrt(3/4))
I get <-2/3, -2/3, 2/3> but this is wrong...
I should be getting <1/2, 0> at max value of 1/2 according to my friend who ceases to explain anything to me.
Any help? Thanks.