Question

find derivative :
$\mathtt \red{\left|\begin{array}{ccc}0 & 1 &s ec \theta \\ \tan \theta & -\sec \theta & \tan \theta \\ 1 & 0 & 1\end{array}\right|}$
Answer with proper explanation.

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Answer 1

$\large\underline{\sf{Solution-}}$

Given determinant is

$\rm :\longmapsto\:\left|\begin{array}{ccc}0 & 1 &s ec \theta \\ \tan \theta & -\sec \theta & \tan \theta \\ 1 & 0 & 1\end{array}\right|$

Let assume that

$\rm :\longmapsto\:y = \left|\begin{array}{ccc}0 & 1 &s ec \theta \\ \tan \theta & -\sec \theta & \tan \theta \\ 1 & 0 & 1\end{array}\right|$

$\red{\rm :\longmapsto\:\boxed{\tt{ OP \: C_3 \: \to \: C_3 - C_1 \: }}}$

We get

$\rm :\longmapsto\:y = \left|\begin{array}{ccc}0 & 1 &sec \theta - 0\\ \tan \theta & -\sec \theta & \tan \theta - tan \theta\\ 1 & 0 & 1 - 1\end{array}\right|$

$\rm :\longmapsto\:y = \left|\begin{array}{ccc}0 & 1 &s ec \theta \\ \tan \theta & -\sec \theta & 0 \\ 1 & 0 & 0\end{array}\right|$

Now, expanding along third column, we get

$\rm :\longmapsto\:y = sec \theta\bigg[0 - ( - sec \theta)\bigg]$

$\rm :\longmapsto\:y = sec^{2} \theta$

On differentiating both sides, we get

$\rm :\longmapsto\:\dfrac{d}{d \theta}y = \dfrac{d}{d \theta}sec^{2} \theta$

We know,

$\red{\rm :\longmapsto\:\boxed{\tt{ \dfrac{d}{dx} {x}^{n} = {nx}^{n - 1} \: }}}$

So, using this, we get

$\rm :\longmapsto\:\dfrac{dy}{d \theta} = 2sec \theta \: \dfrac{d}{d \theta}sec \theta$

$\rm :\longmapsto\:\dfrac{dy}{d \theta} = 2sec \theta \: (sec \theta \: tan \theta)$

$\rm :\longmapsto\:\dfrac{dy}{d \theta} = 2sec^{2} \theta \: tan \theta$

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More to know :-

1. The determinant value remains unaltered if rows and columns are interchanged.

2. The determinant value is 0, if two rows or columns are identical.

3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

4. The determinant value remains unaltered if rows or columns are added or subtracted.

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$