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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY2 Marks
Question
$\text{If y}=\begin{vmatrix}\text{f(x)} & \text{g(x)} & \text{h(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c} \end{vmatrix},\ \text{prove that}\frac{\text{dy}}{\text{dx}}= \begin{vmatrix}\text{f(x)} & \text{g'(x)} & \text{h'(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c}\end{vmatrix}$
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Answer

$ \begin{vmatrix} \text{f(x)} & \text{g(x)} & \text{h(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c} \end{vmatrix}$
$\Rightarrow\ \text{y}=(\text{mc}-\text{nb})\text{f(x)}-(\text{lc}-\text{na})\text{g(x)}+(\text{lb}-\text{ma})\text{h(x)}$
Then, $\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}[(\text{mc}-\text{nb)}\text{f(x)}]-\frac{\text{d}}{\text{dx}}[(\text{lc}-\text{na)}\text{g(x)}]+\frac{\text{d}}{\text{dx}}[(\text{lb}-\text{ma})\text{h(x)}]$
$=(\text{mc}-\text{nb})\text{f}'\text{(x})-(\text{lc}-\text{na})\text{g}'\text{(x)}=(\text{lb}-\text{ma})\text{h}'\text{(x)}$
$= \begin{vmatrix} \text{f}'\text{(x)} & \text{g}'\text{(x)} & \text{h}'\text{(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c} \end{vmatrix}$
Thus, $\frac{\text{dy}}{\text{dx}} = \begin{vmatrix} \text{f}'\text{(x)} & \text{g}'\text{(x)} & \text{h}'\text{(x)} \\ \text{l} & \text{m} & \text{n} \\ \text{a} & \text{b} & \text{c} \end{vmatrix}$

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