Computate the commutator of $[p^n,x]=-i\hbar np^{n-1}$ with $p=-i\hbar \frac{\delta}{\delta x}$ the impulse operator. $\hbar$ stands for $\frac{h}{2\pi}$.
Answer: I do it with induction over $n$. For $n=1$ it is clear (from Lecture). Now $n\to n+1$: For a commutator is $[AB,C]=A[B,C]-[A,C]B$.By applying this on $[p^n,x]$ it follows: $$[p^n,x]=p[p^{n-1},x]-[p,x]p^{n-1}=-p(n-1)i\hbar p^{n-2}+i\hbar p^{n-1}$$ and that is $-i\hbar (n-2)p^{n-1}$.
So what is the mistake?
I suspect your main problem is the commutator formula.
Let's take a look:
$$[AB,C] = (AB)C - C(AB) = ABC - CAB$$
Now, let's look at the right-hand side. The RHS has two terms: $A[B,C]$ and $[A,C]B$.
$$A[B,C] = A(BC - CB) = ABC - ACB$$
$$[A,C]B = (AC-CA)B = ACB - CAB$$
So in fact, the correct formula looks like:
\begin{align} A[B,C]+[A,C]B&= ABC - ACB + ACB - CAB\\ & = ABC - CAB \\ & = [AB,C] \end{align}
So, it looks like the formula you should be using is:
$$[AB,C] = A[B,C]+[A,C]B$$
I believe you can complete the problem from here.