What function satisfies this equation?

(f(x))' = (xf(x))'

I think it is a logarithmic function, but I cannot think of what. Obviously you could define f(x) = 0*x or something similar, but I don't think that is the point.

Question

What function satisfies this equation?

(f(x))' = (xf(x))'

I think it is a logarithmic function, but I cannot think of what. Obviously you could define f(x) = 0*x or something similar, but I don't think that is the point.

anonymous · Answer

f(x) = (1-x)f'(x)

anonymous · Answer

That is interesting, but is there a non-recursive function that can satisfy it?

anonymous · Answer

can you rewrite the equation?

anonymous · Answer

f'(x) = xf'(x)

In other words, the derivative of f(x) must be the same whether it is multiplied by x or not.

anonymous · Answer

any constant function

anonymous · Answer

say f(x) = 1, it will satisfy the equation

anonymous · Answer

Yes, is that truly the only option though? It seems too obvious.

anonymous · Answer

or whenever x = 1, it will be satisfied

anonymous · Answer

Okay, thank you.