- Given that \(k\) is a real non-zero constant and that
$$y = x ^ { 3 } \sin k x$$
use Leibnitz's theorem to show that
$$\frac { \mathrm { d } ^ { 5 } y } { \mathrm {~d} x ^ { 5 } } = \left( k ^ { 2 } x ^ { 2 } + A \right) k ^ { 3 } x \cos k x + B \left( k ^ { 2 } x ^ { 2 } + C \right) k ^ { 2 } \sin k x$$
where \(A\), \(B\) and \(C\) are integers to be determined.