If f, g and h are differentiable functions&#x

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If f, g and h are differentiable functions of x and △ ⎛ ⎜⎜ ⎝ x ⎞ ⎟⎟ ⎠ = ∣ ∣∣∣ ∣ \begin{matrix} f & g & h (x f)' & (x g)' & (x h)' (x^{2} f)'' & (x^{2} g)'' & (x^{2} h)'' \end{matrix} ∣ ∣∣∣ ∣, then △' ⎛ ⎜⎜ ⎝ x ⎞ ⎟⎟ ⎠ is

Options:

A . ∣∣ ∣ ∣∣fghf'g'h'(x3f'')'(x3h'')'(x3h'')'∣∣ ∣ ∣∣

B . ∣∣ ∣ ∣∣fghf'g'h'(x3f'')(x3h'')(x3h'')∣∣ ∣ ∣∣

C . ∣∣ ∣ ∣∣fghf'g'h'(x3f'')''(x3h'')''(x3h'')''∣∣ ∣ ∣∣

D . 0

Answer: Option A
:
A

∣ ∣∣∣ ∣ \begin{matrix} f & g & h x f' + f & x g' + g & x h' + h 4 x f' + 2 f + x^{2} f'' & 4 x g' + 2 g + x^{2} g'' & 4 x h' + 2 h + x^{2} h'' \end{matrix} ∣ ∣∣∣ ∣ Operating R_{2} \to R_{1} - R_{1}; R_{3} \to R_{3} - 4 R_{2} + 2 R_{1} and shifting x of R_{2} to R_{3} △ ⎛ ⎜⎜ ⎝ x ⎞ ⎟⎟ ⎠ = ∣ ∣∣∣ ∣ \begin{matrix} f & g & h f' & g' & h' x^{3} f'' & x^{3} g'' & x^{3} h'' \end{matrix} ∣ ∣∣∣ ∣ \Rightarrow △' ⎛ ⎜⎜ ⎝ x ⎞ ⎟⎟ ⎠ = 0 + 0 + ∣ ∣∣∣ ∣ \begin{matrix} f & g & h f' & g' & h' (x^{3} f'')' & (x^{3} g'')' & (x^{3} h'')' \end{matrix} ∣ ∣∣∣ ∣

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