Differentiable function/R/1 over g 1...g n/Derivation through Induction/Exercise

Let

$g_{1},g_{2},\ldots ,g_{n}\colon \mathbb {R} \longrightarrow \mathbb {R} \setminus \{0\}$

denote differentiable functions. Prove, by induction over ${}n$, the relation

${}{\left({\frac {1}{g_{1}\cdot g_{2}\cdots g_{n}}}\right)}^{\prime }={\frac {-1}{g_{1}\cdot g_{2}\cdots g_{n}}}\cdot {\left({\frac {g_{1}'}{g_{1}}}+{\frac {g_{2}'}{g_{2}}}+\cdots +{\frac {g_{n}'}{g_{n}}}\right)}\,.$

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