Step-by-Step Instructions for finding the derivative WITHOUT using derivative rules?

I need to know how to find the equation for the derivative f'(a), when f(a) = 5x/(1+x^2). If you can show me how to do this in a step-by-step manner without using quotient rule then it would help me out so much. I believe I need to use the (f(x) - f(a) / x-a) formula to solve it, but whenever I plug everything in I get stuck with a complex fraction and I don't know how to simplify it. If you can PLEASE show me how to do this I will be forever grateful. I have spent several completely fruitless hours just trying to solve this one problem. PLEASE HELP!!!! Thank you in advance :-)))