Question

Let $f\left(x\right)=\left|\begin{array}{ccc}\cos (x+x^2)& \sin (x+x^2)& -\cos (x+x^2)\\ \sin (x-x^2)& \cos (x-x^2)& \sin (x-x^2)\\ \sin 2x& 0& \sin \left({2x}^2\right)\end{array}\right|$ find the value of $f^{\prime }\left(0\right)$

• A. $0$
• B. $1$
• C. $2$
• D. $-1$

Answer 1

Answer: (C)

$2$

The given equation is:

$f\left(x\right)=\left|\begin{array}{ccc}\cos (x+x^2)& \sin (x+x^2)& -\cos (x+x^2)\\ \sin (x-x^2)& \cos (x-x^2)& \sin (x-x^2)\\ \sin 2x& 0& \sin \left(2x^2\right)\end{array}\right|$

Differentiating once w.r.t to x we get,

$\Rightarrow f^{\prime }\left(x\right)=\left|\begin{array}{ccc}-\left(\sin (x+x^2)\right).(1+2x)& \left(\cos (x+x^2)\right).(1+2x)& \left(\sin (x+x^2)\right).(1+2x)\\ \sin (x-x^2)& \cos (x-x^2)& \sin (x-x^2)\\ \sin 2x& 0& \sin \left(2x^2\right)\end{array}\right|$ +

$\left|\begin{array}{ccc}\cos (x+x^2)& \sin (x+x^2)& -\cos (x+x^2)\\ \left(\cos (x-x^2)\right).(1-2x)& -\left(\sin (x-x^2)\right).(1-2x)& \left(\cos (x-x^2)\right).(1-2x)\\ \sin 2x& 0& \sin \left(2x^2\right)\end{array}\right|$ +

$\left|\begin{array}{ccc}\cos (x+x^2)& \sin (x+x^2)& -\cos (x+x^2)\\ \sin (x-x^2)& \cos (x-x^2)& \sin (x-x^2)\\ 2\cos 2x& 0& 4x.\cos \left(2x^2\right)\end{array}\right|$

$\Rightarrow f^{\prime }\left(0\right)=\left|\begin{array}{ccc}-\sin 0& \cos 0& \sin 0\\ \sin 0& \cos 0& \sin 0\\ \sin 0& 0& \sin 0\end{array}\right|+\left|\begin{array}{ccc}\cos 0& \sin 0& -\cos 0\\ \cos 0& -\sin 0& \cos 0\\ \sin 0& 0& \sin 0\end{array}\right|+\left|\begin{array}{ccc}\cos 0& \sin 0& -\cos 0\\ \sin 0& \cos 0& \sin 0\\ 2\cos 0& 0& 0\end{array}\right|$

$\Rightarrow f^{\prime }\left(0\right)=\left|\begin{array}{ccc}0& 1& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right|+\left|\begin{array}{ccc}1& 0& -1\\ 1& 0& 1\\ 0& 0& 0\end{array}\right|+\left|\begin{array}{ccc}1& 0& -1\\ 0& 1& 0\\ 2& 0& 0\end{array}\right|$

$\Rightarrow f^{\prime }\left(0\right)=0+0+2$

$\Rightarrow f^{\prime }\left(0\right)=2$