Question about implicit function theorem

[chy1013m1]

if z was solved in terms of x, y, then when we differenciate d (dz/dx) / dx, are we treating z as a constant or still a function of x, y ?

 

[arildno]

Have a bit more precise notation!

Now, for the equation F(x,y,z)=0, the implicit function theorem states that under relatively mild conditions, there exists a function Z(x,y), so that F(x,y,Z(x,y))=0 holds IDENTICALLY in an open region about a solution of F(x,y,z)=0.

We may find the expression for dZ/dx and d(dZ/dx)/dx by differentiatiang our identity with respect to x.

 

[chy1013m1]

so if I was to differenciate d(dZ/dx) / dx for F(x, y, z) = e^z + (x - z) * y - 4 = 0

i'd do: dz/dx = - (dF/dx) / (dF/dz) = - y * (e^z - y)^-1

then d (dz/dx) / dx = y * (e^z-y)^-2 * e^z * (dz/dx)

is that correct ?

 

[arildno]

Assuming your expressions for the derivatives are correct, then you are correct since your method is correct.