Let f(x)  be a function such that f(x+y)=f(x)+f(y)  and f(x)=sinxg(x)  for all x,y∈R .  If  g(x)  is a continuous function such that g(0)=K , then f'(x)  is equal to

f'(x)=limh→0f(x+h)−f(x)h                                  {by definition} ⇒f'(x)=limh→0f(x)+f(h)−f(x)h                   {∵ f(x+y)=f(x)+f(y)} ⇒ f'(x)=limh→0f(h)h=limh→0sinhg(h)h ∴ f'=limh→0sinhhlimh→0g(h)=g(0)=K