structural:scholar_edition, linguistic:paraphrasing
Proof from derivative definition and limit properties

Let h (x) = f (x) g (x). {\displaystyle h(x)={\frac {f(x)}{g(x)}}.} Applying the definition of the derivative and properties of limits gives the following proof, with the term f (x) g (x) {\displaystyle f(x)g(x)} added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:

The limit evaluation lim k → 0 1 g (x + k) g (x) = 1 g (x) 2 {\displaystyle \lim _{k\to 0}{\frac {1}{g(x+k)g(x)}}={\frac {1}{g(x)^{2}}}} is justified by the differentiability of g (x) {\displaystyle g(x)}, implying continuity, which can be expressed as lim k → 0 g (x + k) = g (x) {\displaystyle \lim _{k\to 0}g(x+k)=g(x)}.