EDB — 1DJ

[1DJ] Difficulty:*.Let \(n≥ 1\) be an integer. Let \(I,J\) be open intervals with \(x_ 0∈ I,y_ 0∈ J\). Let then be given \(g:I→ℝ\) and \(f:J→ℝ\) such that \(g(I)⊆ J\), \(f,g\) are \(n-1\) times differentiable in their respective intervals, their \((n-1)\)-th derivative is differentiable in \(x_ 0\) (resp. \(y_ 0\)) and finally \(g(x_ 0)=y_ 0\).

Show that the composite function \(f◦ g\) is differentiable \(n-1\) times in the interval and its derivative \((n-1)\)-th is differentiable in \(x_ 0\).

Then write an explicit formula for the nth derivative \((f◦ g)^{(n)}\) in \(x_ 0\) of the composition of the two functions, (formula that uses derivatives of \(f\) and \(g\)).
(If you can’t find it, read the wikipedia page [ 61 ] ; or, see this presentation: https://drive.google.com/drive/folders/1746bdJ89ZywciaEqvIMlGZ7kKHWVekhb ).