If the derivative is the change of the function at each step, it could be expressed as:
Therefore if $f(x)=c$
$$c+f'(x)=c \implies f'(x)=0$$
This is also correct for $f(x)=cx$
$$cx+f'(x)=c(x+1) \implies f'(x)=c$$
However, it doesn't work for $f(x)=x^2$
$$x^2+f'(x)=(x+1)^2=x^2+2x+1 \implies f'(x)=2x+1$$
Where am I wrong in my reasoning?