Assuming that f and g are functions differentiable at a (though we do not know their formulas). Prove that f +g is differentiable at a using the definition of the derivative.

Answer 1

we know these limits exist as h->0:

(f(a+h)-f(a))/h
(g(a+h)-g(a))/h

sum of limits is thus

(f(a+h)-(a+h)-g(a))/h
= (f(a+h)+g(a+h) - ((a))/h
= ((f+g)(a+h) - (f+g)(a))/h

the limit is d(f+g)/dx at x=a