Let f, g, h be differentiable functions of x. If $Δ=\begin{vmatrix}f&g&h\\(xf)'&(xg)'&(xh)'\\(x^2\, f)'' &(x^2\, g)'' &(x^2\, h)''\end{vmatrix}$ and $Δ'=\begin{vmatrix}f&g&h\\f'&g'&h'\\(x^n\, f'')' &(x^n\, g'')'&(x^n\, h'')'\end{vmatrix}$, then n =

3

We have,

$Δ=\begin{vmatrix}f&g&h\\(xf)'&(xg)'&(xh)'\\(x^2\, f)'' (x^2\, g)'' (x^2\, h)''\end{vmatrix}$

$⇒Δ=\begin{vmatrix}f&g&h\\xf'+f&xg'+g&xh'+h\\x^2\, f''+4xf'+2f &x^2\, g''+4xg'+2g&x^2\, h''+4xh'+2h\end{vmatrix}$

Applying $R_2 → R_2 - R_1$ and $R_3 → R_3 -2 R_1$, we get

$⇒Δ=\begin{vmatrix}f&g&h\\xf'&xg'&xh'\\x^2\, f''+4xf' &x^2\, g''+4xg'&x^2\, h''+4xh'\end{vmatrix}$

$⇒Δ=\begin{vmatrix}f&g&h\\xf'&xg'&xh'\\x^2\, f'' &x^2\, g''&x^2\, h''\end{vmatrix}$  [Applying $R_3 → R_3 -4R_2$]

$⇒Δ=x\begin{vmatrix}f&g&h\\f'&g'&h'\\x^2\, f'' &x^2\, g''&x^2\, h''\end{vmatrix}$ [Taking x common from $R_2$]

$⇒Δ=\begin{vmatrix}f&g&h\\f'&g'&h'\\x^3\, f'' &x^3\, g''&x^3\, h''\end{vmatrix}$  [Multiplying $R_3$ by x]

$∴Δ'=\begin{vmatrix}f'&g'&h'\\f'&g'&h'\\x^3\, f'' &x^3\, g''&x^3\, h''\end{vmatrix}+\begin{vmatrix}f&g&h\\f''&g''&h''\\x^3\, f'' &x^3\, g''&x^3\, h''\end{vmatrix}+\begin{vmatrix}f&g&h\\f'&g'&h'\\(x^3\, f'')' &(x^3\, g'')'&(x^3\, h'')'\end{vmatrix}$

$⇒Δ'=0+0+\begin{vmatrix}f&g&h\\f'&g'&h'\\(x^3\, f'')' &(x^3\, g'')'&(x^3\, h'')'\end{vmatrix}$

$⇒Δ'=\begin{vmatrix}f&g&h\\f'&g'&h'\\(x^3\, f'')' &(x^3\, g'')'&(x^3\, h'')'\end{vmatrix}$

$∴n=3$.