Search any question & find its solution
Question:
Answered & Verified by Expert
If $\mathbf{y}=\left|\begin{array}{ccc}\mathbf{f}(\mathbf{x}) & \mathbf{g}(\mathbf{x}) & \mathbf{h}(\mathbf{x}) \\ \ell & \mathbf{m} & \mathbf{n} \\ \mathbf{a} & \mathbf{b} & \mathbf{c}\end{array}\right|$, prove that
$\frac{\mathbf{d y}}{\mathbf{d x}}=\left|\begin{array}{ccc}
\mathbf{f}^{\prime}(\mathbf{x}) & \mathbf{g}^{\prime}(\mathbf{x}) & \mathrm{h}^{\prime}(\mathbf{x}) \\
\ell & \mathrm{m} & \mathrm{n} \\
\mathbf{a} & \mathbf{b} & \mathrm{c}
\end{array}\right|$
$\frac{\mathbf{d y}}{\mathbf{d x}}=\left|\begin{array}{ccc}
\mathbf{f}^{\prime}(\mathbf{x}) & \mathbf{g}^{\prime}(\mathbf{x}) & \mathrm{h}^{\prime}(\mathbf{x}) \\
\ell & \mathrm{m} & \mathrm{n} \\
\mathbf{a} & \mathbf{b} & \mathrm{c}
\end{array}\right|$
Solution:
1843 Upvotes
Verified Answer
To differentiate a determinant if
$\mathrm{y}=\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ \ell & \mathrm{m} & \mathrm{n} \\ \mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right|$
then $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ \ell & m & n \\ a & b & c\end{array}\right|$
$+\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ 0 & 0 & 0 \\ \mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right|+\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ \ell & \mathrm{m} & \mathrm{n} \\ 0 & 0 & 0\end{array}\right|$
$\therefore \quad \frac{\mathrm{dy}}{\mathrm{dx}}=\left|\begin{array}{ccc}\mathrm{f}^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\ \ell & \mathrm{m} & \mathrm{n} \\ \mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right|$
$\mathrm{y}=\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ \ell & \mathrm{m} & \mathrm{n} \\ \mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right|$
then $\frac{d y}{d x}=\left|\begin{array}{ccc}f^{\prime}(x) & g^{\prime}(x) & h^{\prime}(x) \\ \ell & m & n \\ a & b & c\end{array}\right|$
$+\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ 0 & 0 & 0 \\ \mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right|+\left|\begin{array}{ccc}\mathrm{f}(\mathrm{x}) & \mathrm{g}(\mathrm{x}) & \mathrm{h}(\mathrm{x}) \\ \ell & \mathrm{m} & \mathrm{n} \\ 0 & 0 & 0\end{array}\right|$
$\therefore \quad \frac{\mathrm{dy}}{\mathrm{dx}}=\left|\begin{array}{ccc}\mathrm{f}^{\prime}(\mathrm{x}) & \mathrm{g}^{\prime}(\mathrm{x}) & \mathrm{h}^{\prime}(\mathrm{x}) \\ \ell & \mathrm{m} & \mathrm{n} \\ \mathrm{a} & \mathrm{b} & \mathrm{c}\end{array}\right|$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.