find ( f_{x}(x,y) ) and ( f_{y}(x,y) ). then, find ( f_{x}(3,4) ) and ( f_{y}(4,2) ).\n( f(x,y)=7xy +…

Explanation:

Step1: Find (f_x(x,y))

Differentiate (f(x,y)=7xy + 6y^{4}+2) with respect to (x) (treating (y) as a constant). Using the power rule (\frac{\partial}{\partial x}(ax^{n}y^{m})=nax^{n - 1}y^{m}) (here (n = 1) for the (xy) term), (\frac{\partial}{\partial x}(7xy)=7y), (\frac{\partial}{\partial x}(6y^{4}) = 0) (since (y) is treated as a constant), (\frac{\partial}{\partial x}(2)=0). So (f_x(x,y)=7y).

Step2: Find (f_y(x,y))

Differentiate (f(x,y)=7xy + 6y^{4}+2) with respect to (y) (treating (x) as a constant). Using the power rule (\frac{\partial}{\partial y}(ax^{n}y^{m})=max^{n}y^{m - 1}), (\frac{\partial}{\partial y}(7xy)=7x), (\frac{\partial}{\partial y}(6y^{4})=24y^{3}), (\frac{\partial}{\partial y}(2)=0). So (f_y(x,y)=7x + 24y^{3}).

Step3: Find (f_x(3,4))

Substitute (x = 3) and (y = 4) into (f_x(x,y)). Since (f_x(x,y)=7y), then (f_x(3,4)=7\times4=28).

Step4: Find (f_y(4,2))

Substitute (x = 4) and (y = 2) into (f_y(x,y)). Since (f_y(x,y)=7x + 24y^{3}), then (f_y(4,2)=7\times4+24\times2^{3}=28 + 24\times8=28+192 = 220).

Answer:

(f_x(x,y)=7y), (f_y(x,y)=7x + 24y^{3}), (f_x(3,4)=28), (f_y(4,2)=220)