# Implicit Differentiation

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

I need to find $$u_z$$. 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.

$$f_z + f_uu_z + f_vv_z = 0$$
$$g_z + g_uu_z + g_vv_z = 0$$

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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Tom Mattson
Staff Emeritus