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Mar
7
revised Derivative of inverse function where inverse is known only numerically.
deleted 18 characters in body
Mar
7
answered Derivative of inverse function where inverse is known only numerically.
Mar
7
revised Derivative of inverse function where inverse is known only numerically.
edited title
Mar
7
comment Derivative of inverse function where inverse is known only numerically.
I don't see how it's trivial. The forward derivative is already a third degree polynomial in $x$. The inverse can be computed only numerically so it seems I need a special numerical approximation routine for $f^{-1}(y,a)$ as $y$ depends on $a$ and we don't have the information on how it depends on it...
Mar
7
revised Derivative of inverse function where inverse is known only numerically.
added 63 characters in body
Mar
7
comment Derivative of inverse function where inverse is known only numerically.
But I computed the inverse without problem numerically. I only need to compute it more precisely using some formula. Please note that I also care about partial derivative only (w.r.t. a), not in both variables (the x can be considered fixed). Such function can be plotted and I can see it has inverse (graphically), but I need the formula...
Mar
7
comment Derivative of inverse function where inverse is known only numerically.
One problem is that it is impossible to compute $J^{-1}$ in this case because it is a non-square matrix ($1\times 2$).
Mar
7
comment Derivative of inverse function where inverse is known only numerically.
I found it, thanks.
Mar
7
comment Derivative of inverse function where inverse is known only numerically.
I was looking for that without success (using keyword "multivariable inverse function theorem"). I found something called "implicit function theorem" but I am not sure if this is it nor how to apply it.
Mar
7
asked Derivative of inverse function where inverse is known only numerically.
Feb
25
revised Inverse function theorem for partial derivatives of a vector function
added 31 characters in body
Feb
25
answered Inverse function theorem for partial derivatives of a vector function
Feb
25
revised Inverse function theorem for partial derivatives of a vector function
added 3 characters in body
Feb
25
revised Inverse function theorem for partial derivatives of a vector function
deleted 477 characters in body
Feb
25
revised Inverse function theorem for partial derivatives of a vector function
removed wrong example
Feb
25
suggested approved edit on Inverse function theorem for partial derivatives of a vector function
Feb
25
revised Inverse function theorem for partial derivatives of a vector function
Added example
Feb
25
suggested approved edit on Inverse function theorem for partial derivatives of a vector function
Feb
25
comment Inverse function theorem for partial derivatives of a vector function
So the answer is actually it is not possible to use the inverse function theorem solely for partial derivatives right?
Feb
25
comment Inverse function theorem for partial derivatives of a vector function
Yes. The problem is that I already know values of $F$, $F'_{a}$ and $F^{-1}$ and only need to somehow obtain $F'_a(F^{-1})$ using the reciprocal rule...