1=<k=<n
find $\displaystyle f^{(n)} (x) $ of:
the first :
$\displaystyle
f(x) = |x|^{n + 1}
$
the second is:
$\displaystyle
f(x) = |\sin x|^{n + 1}
$
The first function k-derivate is $\displaystyle f^{k}(x)=\begin{cases}\displaystyle\frac{(n+1)!}{( n+1-k)!}x^{n+1-k},x>0\\ \\ (-1)^{n+1}\displaystyle\frac{(n+1)!}{(n+1-k)!}x^{n+1-k},x<0\\\end{cases}$.
The second involves a lots of fun because you'll have to do the same with a product !