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Question
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Options
\[\frac{5}{4}\]
\[\frac{4}{5}\]
1
0
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Solution
\[ = \frac{1}{2}\sqrt{x} - \frac{2}{\sqrt{x}}\]
\[ = \frac{1}{2} x^\frac{1}{2} - 2 x^{- \frac{1}{2}}\]
Differentiating both sides with respect to x, we get
\[f'\left( x \right) = \frac{1}{2} \times \frac{1}{2} x^\frac{1}{2} - 1 - 2 \times \left( - \frac{1}{2} \right) x^{- \frac{1}{2} - 1} \left[ f\left( x \right) = x^n \Rightarrow f'\left( x \right) = n x^{n - 1} \right]\]
\[ \Rightarrow f'\left( x \right) = \frac{1}{4} x^{- \frac{1}{2}} + x^{- \frac{3}{2}} \]
\[ \therefore f'\left( 1 \right) = \frac{1}{4} \times 1 + 1 = \frac{5}{4}\]
Hence, the correct answer is option (a).
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