11.6#3

I can't figure out how to figure out this equation. What I did first was finding the gradient of the equation, I got gradient of f = <-2xy, -x^2, 3z^2>. Then I solved it at the point, I got gradient of f = <10, -1, 3>. Then I found the unit vector of v. <-3,3,4>/sqrt(34). Finally I dot product to find the directional deriv, which i got -21/sqrt(34). Please help, I don't know what I'm doing wrong.

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As you say: grad(f) = <-2xy, -x^2, 3z^2> . 

This means  grad(f) (5, -1, 1)=<-2(5)(-1), -(5^2), 3(1^2)> = <10, -25, 3> unlike what you say. It looks like you made the mistake of substituting -1 for x in x^2, but really you needed to substitute x=5 into -x^2.

Then since v=<-3,3,4>,  v^ = <-3,3,4>/sqrt(34), as you say, so 

D_v^ f(5, -1, 1)=<10, -25,3>.<-3,3,4>/sqrt(34) =(-30-75+12)/sqrt(34)=-93/sqrt(34)