Why Does Implicit Differentiation Give Contradictory Results for u_z?

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SUMMARY

The discussion centers on the challenges of finding the partial derivative u_z using implicit differentiation on the equations f(x,y,z,u,v) and g(x,y,z,u,v). The user encounters contradictory results when applying the implicit differentiation method and the formula involving f_z, f_u, and g_z, g_u. The confusion arises from uncertainty about the correctness of the formula used for implicit differentiation. The user seeks clarification on their approach and the validity of their results.

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Students and educators in mathematics, particularly those studying calculus and multivariable functions, as well as anyone encountering challenges with implicit differentiation in their coursework.

Icebreaker
[tex]f(x,y,z,u,v)=xe^y+uz-\cos v=2[/tex]
[tex]g(x,y,z,u,v)=u\cos y+x^2v-yz^2=1[/tex]

I need to find [tex]u_z[/tex]. When I try to do it by implicitly differentiating and solving the equation, I get 2 contradictory answers. If I try the formula, i.e.

[tex]f_z + f_uu_z + f_vv_z = 0[/tex]
[tex]g_z + g_uu_z + g_vv_z = 0[/tex]

I get an answer, but I'm not sure if it's right, since it does not equal to the answer I get when I differentiate implicitly. Any help?

Also I'm not entirely sure if my "formula" is right. Maybe this formula is just implicit differentiation, I haven't looked into it.
 
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