Question

Finde tlfx(x,y),fy(x,y),fx(1,3)and fy(_2,4) for the given function .if z=f (x,y)=3x3y2_x2y3+4x+9

Answer 1

SOLUTION

GIVEN

$\sf{f(x, y) = 3 {x}^{3} {y}^{2} - {x}^{2} {y}^{3} + 4x + 9 }$

TO DETERMINE

$\sf{f_x(x, y) \: , \: f_y(x, y) \: , \: f_x(1,3) \: , \: f_y(-2,4)}$

EVALUATION

Here it is given that

$\sf{f(x, y) = 3 {x}^{3} {y}^{2} - {x}^{2} {y}^{3} + 4x + 9 }$

Differentiating partially both sides with respect to x we get

$\sf{f_x (x, y) = 9 {x}^{2} {y}^{2} - 2{x}^{} {y}^{3} + 4}$

Putting x = 1 , y = 3 we get

$\sf{f_x (1, 3) = 9 \times {1}^{2} \times {3}^{2} - 2 \times {1}^{} \times {3}^{3} + 4}$

$\sf{ \implies \: f_x (1, 3) = 81 - 54 + 4}$

$\sf{ \implies \: f_x (1, 3) = 31}$

$\sf{f(x, y) = 3 {x}^{3} {y}^{2} - {x}^{2} {y}^{3} + 4x + 9 }$

Differentiating partially with respect to y both sides we get

$\sf{f_y(x, y) = 6 {x}^{3} {y}^{} - 3 {x}^{2} {y}^{2} }$

Putting x = - 2 & y = 4 we get

$\sf{f_y( - 2, 4) = 6 \times {( - 2)}^{3} \times {4}^{} - 3 \times {( - 2)}^{2} \times {4}^{2} }$

$\sf{ \implies \: f_y( - 2, 4) = - 192 - 192}$

$\sf{ \implies \: f_y( - 2, 4) = - 384}$

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