Maths!
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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.
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?