Differentiation of Determinants

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No need to expand the determinants while differentiating. The formula to find the derivative of determinants is given below:

\begin{array}{l}
y = \left| {\begin{array}{ccccccccccccccc}
{f(x)}&{g(x)}&{h(x)}\\
{p(x)}&{q(x)}&{r(x)}\\
{m(x)}&{n(x)}&{s(x)}
\end{array}} \right|\\
\frac{{dy}}{{dx}} = \left| {\begin{array}{ccccccccccccccc}
{f'(x)}&{g'(x)}&{h'(x)}\\
{p(x)}&{q(x)}&{r(x)}\\
{m(x)}&{n(x)}&{s(x)}
\end{array}} \right| + \left| {\begin{array}{ccccccccccccccc}
{f(x)}&{g(x)}&{h(x)}\\
{p'(x)}&{q'(x)}&{r'(x)}\\
{m(x)}&{n(x)}&{s(x)}
\end{array}} \right| + \left| {\begin{array}{ccccccccccccccc}
{f(x)}&{g(x)}&{h(x)}\\
{p(x)}&{q(x)}&{r(x)}\\
{m'(x)}&{n'(x)}&{s'(x)}
\end{array}} \right|
\end{array}

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