Derivative of a fraction with summation

Question

I have to find the derivative of ${r(x) = \frac{\sum_{i=0}^{n} \frac{w_i}{x-x_i}f_i}{\sum_{i=0}^{n} \frac{w_i}{x-xi}}}$. I know I have to use the chain rule, but I don't know how to apply it in this case (how many functions do we have?), and in addition I don't how how to compute the derivative of this kind of summation.

Answer 1

It could be simpler using the logarithmic derivative $${r(x) = \frac{\sum_{i=0}^{n} \frac{w_i}{x-x_i}f_i}{\sum_{i=0}^{n} \frac{w_i}{x-xi}}}\quad \implies \quad \log[r(x)]=\log\left(\sum_{i=0}^{n} \frac{w_i}{x-x_i}f_i\right)-\log\left(\sum_{i=0}^{n} \frac{w_i}{x-xi} \right)$$ $$\frac{r'(x)}{r(x)}=-\frac{\sum_{i=0}^{n}\frac{w_i}{(x-x_i)^2}f_i} {\sum_{i=0}^{n}\frac{w_i}{x-x_i}f_i}+\frac{\sum_{i=0}^{n}\frac{w_i}{(x-x_i)^2}} {\sum_{i=0}^{n}\frac{w_i}{x-x_i}}$$

Now, use $$r'(x)=r(x) \times \frac{r'(x)}{r(x)}$$ and simplify.