Answer
a) $F^{(3)}(x)=3*f^{(2)}(x)+x*f^{(3)}(x)$
b) $F^{(n)}(x)=n*f^{(n-1)}(x)+x*f^{(n)}(x)$ , $n\geq2
Work Step by Step
In all derivatives, we need to use the product rule.
a) is done on the image below
b) We can prove the equation on the picture by induction. We can see that it checks for $n=2$ from the item (a). Suppose now this equation holds for an arbitrary value $k$, where $k\geq2$:
$F^{(k)}(x)=k*f^{(k-1)}(x)+x*f^{(k)}(x)$
Derivating this equation we will have:
$F^{(k+1)}(x)=k*f^{(k)}(x)+1*f^{(k)}(x)+x*f^{(k+1)}(x)$
$F^{(k+1)}(x)=(k+1)*f^{(k)}(x)+x*f^{(k+1)}(x)$
The last equation is the equation we wanted for $n=k+1$, so it is proved.
