Derivative of the mean-by-mean of a density function?

Question

I have the following mean equation : $$a = \frac{\int_{a^r}^\infty xdF(x)}{1-F(a^r)} $$

Well, I do not understand how this can lead to this :

$$ \frac{da}{da^r} =\frac{dF(a^r)}{1-F(a^r)} (a-a^r) $$

can someone please help?

Thank you.

Answer 1

Consider the antiderivative $G(x)$ such that $G'(x)=xF'(x)$. We can thus write $$ a= \frac{G(\infty)-G(a_r)}{1-F(a_r)} $$ Taking the derivative wrt $a_r$ yields \begin{eqnarray*} \frac{da}{da_r} &=& \frac{-G'(a_r)[1-F(a_r)]+F'(a_r)[G(\infty)-G(a_r)]}{[1-F(a_r)]^2} \\ &=& \frac{-a_rF'(a_r)+aF'(a_r)}{1-F(a_r)} \\ \end{eqnarray*} which is the expected result.