Find $\frac{\partial^2 f_i}{\partial x_j \partial x_k}$ for implicit function $f$

Question

How to find a formula for the second partial derivative of an implicit function

$\frac{\partial^2 f_i}{\partial x_j \partial x_k}$

Could not find any reference to this?

Answer 1

Do you mean you have (I specialise to $i=n+1$) $$F(x_1,x_2,\dots,x_n,f_{n+1}(x_1,x_2,\dots,x_n))=0$$ and would like to express the partial derivatives of $f_{n+1}$ in terms of those of $F$?

If so, let's write $f_{n+1}=f$ for simplicity.

By the Chain Rule we now have on differentiating w.r.t. $x_j$ that $$F_j+F_{n+1}f_j=0$$ which expresses each $f_j$ in terms of the partials of $F$.

Now differentiate this w.r.t. $x_k$, using Chain Rule (and Product Rule) to get $$F_{jk}+F_{j,n+1}f_k+F_{n+1,k}f_j+F_{n+1, n+1} f_j f_k +F_{n+1}f_{jk}=0.$$

It is now tedious algebra to substitute for the $f_j$ and extract an expression for $f_{jk}$.