Question

If $y=\left|\begin{array}{ccc}\sin x& \cos x& \sin x\\ \cos x& -\sin x& x\\ x& 1& 1\end{array}\right|$,find $\frac{dy}{dx}$ at $x=0$

Answer 1

$y=\left|\begin{array}{ccc}\sin x& \cos x& \sin x\\ \cos x& -\sin x& x\\ x& 1& 1\end{array}\right|$
$\frac{dy}{dx}=\left|\begin{array}{ccc}\cos x& -\sin x& \cos x\\ \cos x& -\sin x& x\\ x& 1& 1\end{array}\right|+\left|\begin{array}{ccc}\sin x& \cos x& \sin x\\ -\sin x& -\cos x& 1\\ x& 1& 1\end{array}\right|+\left|\begin{array}{ccc}\sin x& \cos x& \sin x\\ \cos x& -\sin x& x\\ 1& 0& 0\end{array}\right|$
at $x=0$
$\frac{dy}{dx}=\left|\begin{array}{ccc}1& 0& 1\\ 1& 0& 0\\ 0& 1& 1\end{array}\right|+\left|\begin{array}{ccc}0& 1& 0\\ 0& -1& 1\\ 0& 1& 1\end{array}\right|+\left|\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}\right|$
$\therefore \frac{dy}{dx}=1$ at $x=0$.