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If \(y = \begin{vmatrix} f(x) & g(x) & h(x) \\l & m &n \\ a & b & c \end{vmatrix}\), prove that \(\large\frac{dy}{dx}\normalsize = \begin{vmatrix} f'(x) & g'(x) & h'(x) \\l & m &n \\ a & b & c \end{vmatrix}\)

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  • To differentiate a determinant,we differentiate one row at a time keeping other unchange.
If $y=\begin{vmatrix}f(x) & g(x) & h(x)\\l & m & n\\a & b & c\end{vmatrix}$
Then $\large\frac{dy}{dx}$$=\begin{vmatrix}f'(x) & g'(x) & h'(x)\\l & m & n\\a & b & c\end{vmatrix}+\begin{vmatrix}f(x) & g(x) & h(x)\\0& 0 & 0\\a & b & c\end{vmatrix}+\begin{vmatrix}f(x) & g(x) & h(x)\\l & m & n\\0 & 0 & 0\end{vmatrix}$
$\qquad\;\;\;\;=\begin{vmatrix}f'(x) & g'(x) & h'(x)\\l & m & n\\a & b & c\end{vmatrix}$
Hence proved.
answered May 15, 2013 by sreemathi.v
 

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