Given f(x) = g(x) · h(x), where g′(x) and h′(x) both exist, prove the product rule by demonstrating that f′(x) = g(x) · h′(x) + h′(x) · g′(x).

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From (9.2), the derivative of f(x) is: f'(x) = lim f(x+Ax) f(x) - Ax 0 A.x Substituting f(x) = g(x) h(x), f'(x)= lim g(x+Ax) h(x + Ax) - g(x) h(x) Ax-0 A.r Subtracting and adding g(x + Ax) - h(x), f'(x) = lim Ax0 g(x+Ax)h(x+Ax) - g(x + Ax)h(x) + g(x + Ax)h(x) - g(x)h(x) Partially factoring out g(x + Ax) and h(x), . g(x+Ax)[h(x + Ax)-h(x)] + h(x)[g(x + Ax) - g(x)] f'(x) = lim A.x-0 . g(x+Ax)[h(x + Ax) - h(x)] f'(x) = lim Ax0 Ax + lim Ax0 h(x)[g(x+Ax) - g(x)] . h(x+Ax) - h(x) g(x+Ax)g(x) f'(x)=lim_g(x + Ax) lim +lim h(x) lim Ax0 Ax-0 A.x A.x-0 Ax-0 Ax f'(x) = g(x) h'(x) + h(x) g'(x)