Finding the Function f(x) Given Certain Conditions

[root_two]

1. If f(x) be a differentiable function, such that f(x)f(y) + 2 = f(x) + f(y) + f(xy); f'(0) = 0 & f'(1) = 2, then find f(x).

Homework Equations

The Attempt at a Solution

I tried differentiating the given stuff wrt to x and then tried to put in the values given instead of y and find a differential equations by solving which I could get the function.. That didn't work with the stuff tried. Then I went on to substitute y=x and tried to get a differential equation. But I was unable to write f'(x^2)2x in terms of dy/dx.. So, I couldn't go ahead. Also, was it mathematically correct to put in x=y to solve the question?

 

[SammyS]

1. If f(x) be a differentiable function, such that f(x)f(y) + 2 = f(x) + f(y) + f(xy); f'(0) = 0 & f'(1) = 2, then find f(x).

Homework Equations

The Attempt at a Solution

I tried differentiating the given stuff wrt to x and then tried to put in the values given instead of y and find a differential equations by solving which I could get the function.. That didn't work with the stuff tried. Then I went on to substitute y=x and tried to get a differential equation. But I was unable to write f'(x^2)2x in terms of dy/dx.. So, I couldn't go ahead. Also, was it mathematically correct to put in x=y to solve the question?

What did you get when you differentiated?

Yes, it was mathematically correct to put in x=y to help solve this.

 

[root_two]

Well actually on differentiating I am getting this:
2f(x)f'(x) = 2f'(x) + f'(x^2)2x

Now I can put f'(x) as dy/dx but what do I do with f'(x^2)2x??