Question

Proof of the product rule: Suppose that each of \(g\) and \(h\) is a function with domain all numbers so that each of \(g'\) and \(h'\) have domain all numbers. Suppose also that \(f\) is the function so that \(f (x) = g (x) h (x)\) for all numbers \(x\) (that is, \(f = g h\)). Then

\(\frac{f (p) - f (a)}{p - a} = g (\) \() \frac{h (p) - h (a)}{p - a} + h (\) \() \frac{g (p) - g (a)}{p - a}\)

As \(p \rightarrow a\), the first quotient on the right goes to \(\left. \frac{d}{d x} \right|_{x = a}\)\((x)\)

As \(p \rightarrow a\), the second quotient on the right goes to \(\left. \frac{d}{d x} \