Proving the sum rule

Question

Show that if each of f and g is a function and S(x) = f(x) + g(x), then S’(x) = f’(x) + g’(x).

Would I use the limit formula

lim_x->a [f(x)+g(x)] = lim_x->a f(x) + lim _x->a g(x).

I have the option of presenting this problem in front of the class but I’m worried I will not be able to explain it because I still don’t understand how to work the problem out

Answer 1

I would use the

lim_h->0(f(x+h)-f(x))/h definition of the derivative

S'(x)=lim_h->0(S(x+h)-S(x))/h

=lim_h->0((f(x+h)+g(x+h))-(f(x)+g(x)))/h

Distribute the negative

=lim_h->0(f(x+h)+g(x+h)-f(x)-g(x))/h

group f and g terms together and split into two fractions

=lim_h->0((f(x+h)-f(x))/h+(g(x+h)-g(x))/h)

Split into two limits

=lim_h->0(f(x+h)-f(x))/h+lim_h->0(g(x+h)-g(x))/h

S'(x)=f'(x)+g'(x)