for the function ( f(x,y)=x^{2}e^{3xy} ), find ( f_{x},f_{y},f_{x}(1,-1) ), and ( f_{y}(2,4) ).\n(…

for the function ( f(x,y)=x^{2}e^{3xy} ), find ( f_{x},f_{y},f_{x}(1,-1) ), and ( f_{y}(2,4) ).\n( f_{x}=square )\n( f_{y}=square )\n( f_{x}(1,-1)=square ) (type an exact answer.)\n( f_{y}(2,4)=square ) (type an exact answer.)

for the function ( f(x,y)=x^{2}e^{3xy} ), find ( f_{x},f_{y},f_{x}(1,-1) ), and ( f_{y}(2,4) ).\n( f_{x}=square )\n( f_{y}=square )\n( f_{x}(1,-1)=square ) (type an exact answer.)\n( f_{y}(2,4)=square ) (type an exact answer.)

Answer

Explanation:

Step1: Find the partial derivative (f_x)

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = x^{2}), (v=e^{3xy}). (u^\prime = 2x), (v^\prime=e^{3xy}\cdot3y) (f_x=\frac{\partial f}{\partial x}=2x e^{3xy}+x^{2}\cdot3y e^{3xy}=x e^{3xy}(2 + 3xy))

Step2: Calculate (f_x(1,-1))

Substitute (x = 1), (y=-1) into (f_x): (f_x(1,-1)=1\times e^{3\times1\times(-1)}(2+3\times1\times(-1))=e^{- 3}(2 - 3)=-e^{-3})

Step3: Find the partial derivative (f_y)

(f_y=\frac{\partial f}{\partial y}=x^{2}\cdot e^{3xy}\cdot3x=3x^{3}e^{3xy})

Step4: Calculate (f_y(2,4))

Substitute (x = 2), (y = 4) into (f_y): (f_y(2,4)=3\times2^{3}\times e^{3\times2\times4}=24e^{24})

Answer:

(f_x(1,-1)=-e^{-3}), (f_y(2,4)=24e^{24})