When we compute derivatives on functional equations such as Cauchy's functional equation, is it possible to say we do partial differentiation?
For example, in Cauchy's functional equation, $$f(x+y)=f(x)+f(y), f:\mathbb{R}\to\mathbb{R}$$ Let's hold $y$ constant $y_0$,$$f(x+y_0)=f(x)+f(y_0)$$ If we differentiate both sides of the equation, we get the next equation. $$ f'(x+y_0)=f'(x)$$ Could we say we computed partial derivative of the $f_x$ because of $y$ held constant?
Since $f$ is a map from $\mathbb R$ into $\mathbb R$, it is weird to talk about partial derivatives here; there is just the derivative of $f$. On the other hand, you proved correctly that, if $f$ is a differentiable solution of Cauchy's functional equation, the$$(\forall x,y_0\in\mathbb R):f'(x+y_0)=f'(x).$$In other words, you have proved that $f'$ is constant. That's not unexpected, since every continuous solution of Cauchy's functional equation is of the form $x\mapsto cx$, for some constant $c$.