EDB — 1DG

[1DG]5 Let \(n≥ 1\) be an integer. Let \(I\) be an open interval and \(x_ 0∈ I\), let \(f,g:I→ℝ\) be functions \(n-1\) times differentiable in the interval, and whose \((n-1)\)-th derivative is differentiable in \(x_ 0\).

Show that the product \(fg\) is differentiable \(n-1\) times in the interval, and its \((n-1)\)-th derivative is differentiable in \(x_ 0\). Write an explicit formula for the n-th derivative \((fg)^{(n)}\) in \(x_ 0\) of the product of the two functions, (formula that uses derivatives of only \(f\) and only \(g\)).

(If you don’t find it, look in Wikipedia at the General Leibniz rule [ 54 ] ) .