Given f(x) = g(x) + h(x), where g(x) and h(x) are both differentiable functions, prove the rule of sums by demonstrating that f′(x) = g′(x) + h′(x).

From (9.2) the derivative of f(x) is

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Substituting f(x) = g(x) + h(x), f'(x) = lim f(x+Ax) f(x) A.x-0 Ax f'(x)= lim Ax-0 [g(x+Ax)+h(x + Ax)]-[g(x) + h(x)] A.x Rearranging terms, g(x+Ax)g(x) + h(x + Ax) - h(x) f'(x) = lim Ax0 A.r Separating terms, and taking the limits, g(x+Ax)g(x) h(x+Ax) h(x) - f'(x) = lim + .-0 . f'(x) = lim g(x+Ax)g(x) Ax0 + lim Ax 0 A.x h(x+Ax) h(x) Ax f'(x) = g'(x)+h'(x)