Calc
posted by Dani 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 have no idea how to solve this problem. Help, please? Thank you.

From point (2,1) to (1,3):
u = <12,31> = <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 = <52,51> = <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