Is anybody here good at calculus? I need to prove the product rule f(x)*g(x)=F(x),

F'(x)=g'(x)*f(x)+f'(x)+g(x), using first principles. Help?

Write the derivative of F(x) as the limit as r (or whatever other constant you feel like using) approaches x of (F(r)-f(x))/(r-x).

 This is the same as the limit as r approaches x of (f(r)g(r)-f(x)g(x))/(r-x).

 Note: (f(r)g(r)-f(x)g(x)) can be rewritten as f(x)(g(r)-g(x))-g(r)(f(r)-f(x))

Rewrite your limit using that, and separate it out. You should get the desired result.