Let X be the set of differentiable functions in real line. Find an
infinity set $H$ in X such that
$ f,g\in H ~~\rightarrow (fg)'=f+g $
Let X be the set of differentiable functions in real line. Find an
infinity set $H$ in X such that
$ f,g\in H ~~\rightarrow (fg)'=f+g $
We can find these functions by noting that the condition has to be fulfilled for $f=g=h$, which leads to a simple differential equation:
$ \begin{align} (hh)'&=h+h\;, \\ 2hh'&=2h\;. \end{align} $
Unless $h=0$, this implies $h'=1$, with solutions $h=x+c$, and indeed
$ ((x+a)(x+b))'=(x+a)+(x+b) $
for all $a,b\in\mathbb R$.
With hindsight, another way to find this solution (but without showing that it's unique) would have been to look at the product rule, $(fg)'=f'g+fg'$, and note that it yields $f+g$ if $f'=g'=1$.