Question

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 have no idea how to solve this problem. Help, please? Thank you.

Answer 1

From point (2,1) to (1,3):
u = <1-2,3-1> = <-1,2>
Note that vector u above is not a unit vector, so we need to make this a unit vector by dividing it with its magnitude.
u = <-1,2>/sqrt(5)

<dz/dx,dz/dy>•<-1/sqrt(5),2/sqrt(5)> = -2/sqrt(5)
-dz/dx + 2dz/dy = -2. ..(1)

From point (2,1) to (5,5):
v = <5-2,5-1> = <3,4>
Again, this is not a unit vector.
v = <3,4>/5 = <3/5,4/5>

<dz/dx,dz/dy>•<3/5,4/5> = 1
3dz/dx + 4dz/dy = 5. ..(2)

Using elimination and substition to both equations (1) and (2), you'll get the value for dz/dx and dz/dy