Question

Let $f\left(x\right)=\left|\begin{array}{ccc}cosx& sinx& cosx\\ cox2x& sin2x& 2cos2x\\ cos3x& sin3x& 3cos3x\end{array}\right|$, then $f^{\prime }\left(\frac{\pi }{2}\right)$ is equal to

• A. 8
• B. 6
• C. 4
• D. 2

Answer 1

The correct option is C 4

$f^{\prime }\left(x\right)=\left|\begin{array}{ccc}-sinx& sinx& cosx\\ -2sin2x& sin2x& 2cos2x\\ -3sin3x& sin3x& 3cos3x\end{array}\right|+\left|\begin{array}{ccc}cosx& cosx& cosx\\ cos2x& 2cos2x& 2cos2x\\ cos3x& 3cos3x& 3cos3x\end{array}\right|+\left|\begin{array}{ccc}cosx& sinx& -sinx\\ cos2x& sin2x& -4sin2x\\ cos3x& sin3x& -9sin3x\end{array}\right|$
$f^{\prime }\left(\frac{\pi }{2}\right)=\left|\begin{array}{ccc}-1& 1& 0\\ 0& 0& -2\\ 3& -1& 0\end{array}\right|+0+\left|\begin{array}{ccc}0& 1& -1\\ -1& 0& 0\\ 0& -1& 9\end{array}\right|=2(1-3)+0+1(9-1)=-4+8=4$