Lauren B. answered • 10/02/19

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)