Mark the Correct Alternative in of the Following: If Y = 1 + 1 X 2 1 − 1 X 2 Then D Y D X = - Mathematics

MCQ

Mark the correct alternative in  of the following:

If $y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}$ then $\frac{dy}{dx} =$

Options

• $- \frac{4x}{\left( x^2 - 1 \right)^2}$

• $- \frac{4x}{x^2 - 1}$

• $\frac{1 - x^2}{4x}$

• $\frac{4x}{x^2 - 1}$

Solution

$y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}$
$= \frac{x^2 + 1}{x^2 - 1}$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx} = \frac{\left( x^2 - 1 \right) \times \frac{d}{dx}\left( x^2 + 1 \right) - \left( x^2 + 1 \right) \times \frac{d}{dx}\left( x^2 - 1 \right)}{\left( x^2 - 1 \right)^2} \left( \text{ Quotient rule } \right)$
$= \frac{\left( x^2 - 1 \right) \times \left( 2x + 0 \right) - \left( x^2 + 1 \right) \times \left( 2x - 0 \right)}{\left( x^2 - 1 \right)^2}$
$= \frac{2 x^3 - 2x - 2 x^3 - 2x}{\left( x^2 - 1 \right)^2}$
$= \frac{- 4x}{\left( x^2 - 1 \right)^2}$

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 5 | Page 48