f '(x)=a*f(x)
or, d(f(x))/f(x)=a*dx
or,integration[d(f(x))/f(x)]=integration(a*dx)
or,ln(f(x))=ax+c
or,f(x)=e^(ax+c)
Suppose f:R->R is differentiable everywhere.
Prove that if f′(x) = af(x) for all x, then f(x) = Aexp(ax) for some constant
A.
I've done a lot of fiddling around with this and seem be getting a circular argument. Any help would be greatly appreciated.
Thanks.