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Mark the Correct Alternative in of the Following: If F ( X ) = X − 4 2 √ X - Mathematics

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MCQ

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

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

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).

Concept: The Concept of Derivative - Algebra of Derivative of Functions
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Q 2 | Page 47

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