Question

How do we derive the sum rule in differentiation? (ie. (u+v)' = u' + v')

Answer 1

It follows from the definition of the derivative and basic properties of arithmetic. Let f(x) and g(x) be functions. Their derivatives, if the following limits exist, are

`\displaystyle f'(x) = \lim_(h\to0)\frac{f(x+h)-f(x)}h\text{ and }g'(x)\lim_(h\to0)\frac{g(x+h)-g(x)}h`

The derivative of f(x) + g(x) is then

`\displaystyle \big(f(x)+g(x)\big)' = \lim_(h\to0)\big(f(x)+g(x)\big) \\\\ \big(f(x)+g(x)\big)' = \lim_(h\to0)\frac{\big(f(x+h)+g(x+h)\big)-\big(f(x)+g(x)\big)}h \\\\ \big(f(x)+g(x)\big)' = \lim_(h\to0)\frac{\big(f(x+h)-f(x)\big)+\big(g(x+h)-g(x)\big)}h \\\\ \big(f(x)+g(x)\big)' = \lim_(h\to0)\frac{f(x+h)-f(x)}h+\lim_(h\to0)\frac{g(x+h)-g(x)}h \\\\ \big(f(x)+g(x)\big)' = f'(x) + g'(x)`