problem

Hi Dr. Taylor,

 I need help answering this question. I dont know how to answer without an equation. Thank you.

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OK, first of all: at the bottom of every webwork page is a little button that says "Email Instructor".  If are having trouble with a problem you push that button and it will send me all of the info from the work you have been doing.
The equation you need here is the definition of directional derivative:
D_u f = ∇f . u = <∂f/∂x, ∂f/∂y>. u  for a unit vector u.  You are given a vector <4,6> which has unit vector <4/√52, 6/√52> and a vector <3,7> that has unit vector <3/√58,7/√58>.  This means that the directional derivatives given in the problem satisfy
6/√52 =  ∇f . <4/√52, 6/√52> or 6 =  ∇f . <4, 6> and
6/√58 =  ∇f . <3/√58, 7/√58> or 6 =  ∇f . <3, 7>.
These amount to the two equations
4 ∂f/∂x + 6 ∂f/∂y = 6
3 ∂f/∂x + 7 ∂f/∂y = 6
which you can solve for the two unknowns  ∂f/∂x, ∂f/∂y