State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . Verify this theorem for the functions f and g, defined by

A necessary condition for the functional dependence of two differentiable functions \( f \) and \( g \) on an open subset \( D \) of \( \mathbb{R}^n \) is that the gradients of \( f \) and \( g \) must be proportional, i.e., there exists a scalar function \( _____ ____ ______ _______ ________ _______ __________ ____.
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