If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1\end{array}\right|$…
If $f(x)=\left|\begin{array}{ccc}\cos x & x & 1 \\ 2 \sin x & x^2 & 2 x \\ \tan x & x & 1\end{array}\right|$, then the value of $f^{\prime}(x)$ at $x=0$ is equal to
-1
1
2
0
Solution
$
\begin{aligned}
& \text { Given, } f(x)=\left|\begin{array}{ccc}
\cos x & x & 1 \\
2 \sin x & x^2 & 2 x \\
\tan x & x & 1
\end{array}\right| \\
& \therefore \quad f^{\prime}(x)=\left|\begin{array}{ccc}
-\sin x & 1 & 0 \\
2 \sin x & x^2 & 2 x \\
\tan x & x & 1
\end{array}\right|+\left|\begin{array}{ccc}
\cos x & x & 1 \\
2 \cos x & 2 x & 2 \\
\tan x & x & 1
\end{array}\right| \\
& +\left|\begin{array}{ccc}
\cos x & x & 1 \\
2 \sin x & x^2 & 2 x \\
\sec ^2 x & 1 & 0
\end{array}\right| \\
&
\end{aligned}
$
So, at $x=0, f^{\prime}(x)=\left|\begin{array}{ccc}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{array}\right|+\left|\begin{array}{lll}1 & 0 & 1 \\ 2 & 0 & 2 \\ 0 & 0 & 1\end{array}\right|$
$
+\left|\begin{array}{lll}
1 & 0 & 1 \\
0 & 0 & 0 \\
1 & 1 & 0
\end{array}\right|=0
$