Let f(x) and g(x) be functions s.t.
f(a+b)=f(a)g(b)+g(a)f(b) for all a,b in R
Given that f(0)=0,f'(0)=1,g(0)=1 and g'(0)=0, show that
f'(x)=g(x) for all x in R
Would appreciate any hints that could be given.
hmm, I tried this ("tried" implies that it fails but you might be able to work some magic on it):
I'm clearly not getting g(x). I think this method is the way to go, except it doesn't use the information f'(0)=1 and g'(0)=0.
I haven't tried induction yet but it works for x=0. This method might take you somewhere.
O.K. Differentiating implies that you have assumed that the function is differentiable for all x whereas you have been provided values of the derivatives at only 2 points viz. x=a and x=b .That is why you can use only the definition of the derivative.By doing this we are actually checking whether derivatives exist for all x or not if they exist at the given points.
However, had it been given that the function is differentiable I would not have any qualms by proceeding in the way you have but with a slight difference.
Differentiating with respect to treating as a constant