Prove that it is impossible to write x = f(x)g(x) where f and g are differentiable and f(0) = g(0) = 0.

Hint: Differentiate

I normally am giving math help around here but a student in my tutorial posed this question and I'm unable to help him out. Any help would be appreciated. Thank you. :)

Differentiate both sides and you get

1 = f'(x)g(x) + f(x)g'(x)

Thus, 1 = f'(0)g(0) + f(0)g'(0) = 0 + 0 = 0, a contradiction.

differentiate the expression with respect to x:

1=f '(x)*g(x) + f(x)*g'(x).

Now if f(0)=g(0)=0 then:

1=f '(0)*0 + 0* g'(0) = 0.

but 1 = 0 makes no sense so the assumption that such functions could exist must be false.