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If $y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$ prove that $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right|$
If $y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$ prove that $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right|$
CBSE | Class 12 | Continuity and Differentiability | Maths | Miscellaneous | NCERT
If $y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$ prove that $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ l & m & n \\ a & b & c\end{array}\right|$

Solution:

The provided expression is

$y=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$

Now applying the derivative:

$\frac{d y}{d x}=\left|\begin{array}{ccc}\frac{d}{d x} f(x) & \frac{d}{d x} g(x) & \frac{d}{d x} h(x) \\ l & m & n \\ a & b & c\end{array}\right|+\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ 0 & 0 & 0 \\ a & b & c\end{array}\right|+y$$=\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\ l & m & n \\ 0 & 0 & 0\end{array}\right|$

$=\left|\begin{array}{ccc}1 & 1 & 1 \\ f(x) & g(x) & h(x) \\ l & m & n \\ a & b & c\end{array}\right|$

i

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