Prove by induction $f(x)=x\ln x,x>0$ and $n\in N$. Prove if $n\ge 3$, then $f^{(n)}(x)=(-1)^n[(n-2)!/x^{n-1}]$

Question

$f(x)=x\ln x, x>0$ and $n\in N$. Let $f^{(n)}(x)$ denote the $n$th derivative. Prove if $n\ge 3$, then $f^{(n)}(x)=(-1)^n[(n-2)!/x^{n-1}]$

I don't understand the solution after it does the base case of $n=3$ then it assumes this statement is true for $n=m$ and that $f^{(m)}(x)=(-1)^m[(m-2)!/x^{m-1}]$

Thus,
$f^{(m+1)}(x)=df^{(m)}/dx=(-1)^m(1-m)[(m-2)!/x^m]=(-1)^{m+1}[(m-1)!/x^m]$

I'm confused how they got $(-1)^{m+1}$ on the right.

Answer 1

They factored $-1$ from the $(1-m)$ to get $(-1)(m-1)$ and then used the fact that $(m-1)(m-2)! = (m-1)!$