Directional derivative

[Rumor]

Here's the question I'm stuck on:

"The directional derivative of z = f(x,y) at (2,1) in the direction toward the point (1,3) is -2/sqrt(5), and the directional derivative in the direction toward the point (5,5) is 1. Compute dz/dx and dz/dy at (2,1)."

I'm not even sure how to go about solving this problem. Could someone help me with this?

 

[tkhunny]

You'll need a UNIT vector in the desired direction.

(2,1) toward (1,3) gives (-1/sqrt(5),2/sqrt(5))

If (u,v) is the gradient, we have (u,v)[dot](-1/sqrt(5),2/sqrt(5)) = -u/sqrt(5) + 2v/sqrt(5)) = -2/sqrt(5)

Similarly

(2,1) toward (5,5) gives (3/5,4/5)

If (u,v) still is the gradient, we have (u,v)[dot](3/5,4/5) = 3u/5 + 4v/5 = 1

You're all set. The rest is elementary algebra. Sorry, I did all the hard work. Find another example and work it until you get it.

 

[Rumor]

Ah, I see. I'll work through some more until I get the hang of it. Thanks.