Suppose Du f(1,2)=-5 and Dv f(1,2)=10, where u=3/5i-4/5j and v=4/5i+3/5j. Find fx (1,2) and fy(1,2)The u in Du f and the x and y in fx & fy are supposed to be subscript!Thanks!

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use directional derivatives to find partial derivatives `f_x(1,2)`  and `f_y(1,2) `  such that:

`D_u f(1,2) = (3/5)*f_x(1,2)- (4/5)*f_y(1,2)`

`D_v f(1,2) = (4/5)*f_x(1,2)+ (3/5)*f_y(1,2)`

The problem provides the information `D_u f(1,2) = -5, D_v f(1,2) = 10,`  hence you need to substitute -5 for `D_u f(1,2)`  and 10 for `D_v f(1,2)`  in equations above such that:

`(3/5)*f_x(1,2) - (4/5)*f_y(1,2) = -5`

`(4/5)*f_x(1,2) + (3/5)*f_y(1,2) = 10`

Notice that `f_x(1,2) ` and `f_y(1,2)`  need to be determined, hence you should solve the system of simultaneous equations such that you need to multiply the first equation by `3/5`  and the second equation by `4/5` :

`(9/5)*f_x(1,2)- (12/5)*f_y(1,2) + (16/5)*f_x(1,2) + (12/5)*f_y(1,2) = 10= -15/5 + 40/5`

`(25/5)*f_x(1,2) = 25/5 =gt f_x(1,2) = 1`

You need to substitute 1 for `f_x(1,2)`  in equation `(4/5)*f_x(1,2) + (3/5)*f_y(1,2) = 10`  such that:

`(4/5) + (3/5)*f_y(1,2) = 10 =gt (3/5)*f_y(1,2) = 10 - 4/5`

`(3/5)*f_y(1,2) = 46/5 =gt f_y(1,2) = 46/3`

Hence, evaluating partial derivatives `f_x(1,2)`  and `f_y(1,2)`  under given conditions yields `f_x(1,2) = 1` and `f_y(1,2) = 46/3.`

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