Cfrgtkky

I encountered a question given $theta$ in a random sample of size $n$ and also

$$-ntimes Eleft[frac{partial^2}{partial theta^2}times ln f(X)right]$$

, where $f(x)$ is the p.d.f. at $x$, provided that the extremes of the region for which $f(x) neq0$ do not depend on $theta$. The derivation of this formula take the following steps:

(a) Differentiating the expressions on both sides of
$int f(x),dx=1$
with respect to $theta$, show that

$$int frac{partial}{partialtheta} ln f(x)times f(x),dx=0$$

by interchanging the order of integration and differentiation.

Now I am having trouble understanding what this question was asking me to do. I used

$$int f(x),dx=1$$

and then,

$$int frac{partial}{partialtheta} f(x), dx = 0$$

But did not know how to proceed to get the answer.

In case anyone want to know, the (b) is:
Differentiating again with respect to $theta$, show that.

$$Eleft[left(frac{partial}{partialtheta} ln f(X)right)^2right]=-Eleft[frac{partial^2}{partialtheta^2} ln f(X)right]$$

Thank you very much for your reading and any assistance would be appreciated!! Thanks!