Briefly I am on a calculus course and right now we are learning Brownian Motion, its properties and proofs. And I have a question such as below in my study set that I cant find a solution. I would apreciate any help.
Show that,
$d(W(t)^n) = \frac{n(n-1)}{2}W(t)^{(n-2)}dt + nW(t)^{(n-1)}dW(t)$
Let $f(x,t) = x^n$, for $n \geq 2$ $$\frac{\partial f}{\partial t} = 0,\quad \frac{\partial f}{\partial x} = nx^{n-1},\quad \frac{\partial^2 f}{\partial x^2} = n(n-1)x^{n-2}$$
Let also $X_t = W_t$, and apply Ito's lemma to $f(X_t)$ $$df(X_t) = d(W_t^n)$$
Can you take it from here ?