Find Z(subx) and z(suby)

Question

Let e^xyz = 2. Find z_x and z_y in 3 ways and check for agreement. 

 

a. use result of a previous exercise (i.e. Assume that F(x,y,z(x,y))=0 implicitly defines z as a differentiable function of x and y.Extend this theorem [Let F be differentiable on its domain and suppose that F(x, y)=0 defines y as a differentiable function of x. Provided F_y ≠0, dy/dx=-F_x/F_y .] to show that ∂z/∂x=-F_x/F_z and ∂z/∂y=-F_y/F_z.)

 

b. Take logarithms of both sides and differentiate xyz=ln2

 

c. solve for z and differentiate z=ln2/(xy).

 

Thank you

Answer 1

e^xyz=2,   

xyz=ln2,

 

i) z=ln2/(xy)

   z_x=-(ln2/x²y)

   z_y=-(ln2/xy²)

 

iii) d(e^xyz)=0

     d(e^xyz)_xdx+d(e^xyz)_ydy+d(e^xyz)_zdz=0

     (yz+xyz'_x)dx+(xz+xyzz'_y)dy=0

yz+xyz'_x=0, yz+xyz'_y=0

z'_x=-z/x

z'_x=(-ln2/(xy))/x=-ln2/(x²y).  Similar to this z'_y=-ln2/(xy²).

ii) e^xyz=2

(e^xyz)'_x=0

(yz+xyz'_x)e^xyz=0 (y≠0, divide by y)

xz'_x=z

z'_x=-z/x=-ln2/x²y

Similar

z'_y=-z/y=-ln2/xy²

 

a) F(x, y, z(x,y))=e^xyz-2

DF=F_xdx+F_ydy+F_z((∂z/∂x)dx+(∂z/∂y)dy)=0

[F_x+F_z(∂z/∂x)dx]+[F_y+F_z(∂z/∂y)dy]=0

F_x+F_z(∂z/∂x)=0 ⇒ ∂z/∂x=-F_x/F_z

F_y+F_z(∂z/∂y)=0 ⇒ ∂z/∂x=-F_y/F_z

You can check these partial derivarives by taking derivtaive of 

F(x,y,z(x,y))=e^xyz-2.

 

Parts b) and c) are already done above.