So I want to take the first and second derivatives of a function g(Z) which is made up of several terms, one of which is
where Z and H are our variables.
Taking the derivative of this, it seems to me that the integral would evaluate to zero in applying the upper bound according to the Fundamental Theorem of Calculus; we would receive
(Z – Z) = 0, taking our solution to 0 – Z*F(0).
Am I on the right track or is there special consideration that needs to be paid to the (Z-H) term as we evaluate the derivative? Since our overall function g(Z) is in terms of Z, it seems the goal would be to eliminate H from the expression, and integrating by parts doesn't look like it will help there.
Best Answer
HINT:
I would say
$\displaystyle g(z) = \int_{0}^{z}(z-h)f(h)\,dh=z\int_{0}^{z}f(h)\,dh-\int_{0}^{z}hf(h)\,dh$
so $\displaystyle g'(z) = \left (z\int_{0}^{z}f(h)\,dh\right)'-\left (\int_{0}^{z}hf(h)\,dh\right )'=\cdots$