Experts please help. Thank you!

Question

Calculate az/ax and az/ay using implicit differentiation. Leave your answers in terms of x,y, and z.

x^9+zsin(xyz)=0

AZ/AX=?

AZ/AY=?

Please answer AZ/AX and AZ/AY.

Answer 1

Let F ( x, y, z ) = x⁹ + z sin (x y z).

Then by the Implicit Function Theorem ∂z/∂x = - [ ∂F/∂x] / [∂F/∂z] and ∂z/∂y = - [ ∂F/∂y] / [∂F/∂z]

∂z/∂x = - [ ∂F/∂x] / [∂F/∂z]= [ 9 x⁸ +y z² cos (x y z )] / [ x y z cos ( x y z) - sin (x y z) ] and

∂z/∂y = - [ ∂F/∂y] / [∂F/∂z] =[ x z² cos ( x y z) ] / [ x y z cos ( x y z) - sin (x y z) ] .