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

Answer 1

\[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
\[ = \frac{1}{2}\sqrt{x} - \frac{2}{\sqrt{x}}\]
\[ = \frac{1}{2} x^\frac{1}{2} - 2 x^{- \frac{1}{2}}\]

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).