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Question: Implicitdiff(P(u,v),v,u,u) at a point u0, v0 such that its direct substitution leads to a quotient of type 0/0

Question:Implicitdiff(P(u,v),v,u,u) at a point u0, v0 such that its direct substitution leads to a quotient of type 0/0

Jesús Guillera 75 Maple
implicit-differentation substitution
November 06 2024
1

Let P(u,v) = -u⁴+88u³v-146u²v²+88uv³-v⁴+2u²+40uv+2v²-1, with P(u,v)=0,

and u0 = sqrt(2)/6 i,  v0 = -sqrt(2)/6 i. We see that P(u0,v0)=0.

Direct substitution of implicitdiff(P(u,v),v,u) at u0, v0 leads to 0/0.

For obtaining the true value of dv/du at (u0,v0) we have written in Maple:

solve(limit(subs(v=v0+k*(u-u0), implicitdiff(P(u,v),v,u)), u=u0)-k, k);

There are two solutions: 49/113 - 72*sqrt(2)/113*I, and 49/113 + 72*sqrt(2)/113*I.

However we were unable to determine the true value of d²v/du² at (u0,v0).

QUESTION:  How to determine it?
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