Density function of $\max(X_1,\dots,X_n)$.

Density function of $\max(X_1,\dots,X_n)$.

I'm making this statistics exercise and I'm not sure about my solution.
Find the density function of $Y=\max(X_1,\dots,X_n)$ if they are all i.i.d.
This was my take on this question: $F_Y(a)=P(X_1 \leq a, \dots, X_n \leq
a)$. Using that they are independent this gives $F_Y(a)=P(X_1 \leq a)
\cdot P(X_2 \leq a) \dots P(X_n \leq a)= (P(X_1 \leq a))^n=
(F_{X_1}(a))^n$. So the density function is $f_Y(x)= \frac{\partial
F_Y(x)}{\partial x}= n\cdot f_{X_1}(x) \cdot F_{X_1}^{n-1}(x)$.
What do you think of this argument?
How would you calculate $E[Y]$?
Thanks!