Question

y =x² + 3 / x² - 5 then dy / dx =

a) 16x / ( x²- 5 )² b) -16x / ( x²- 5 )²

Answer 1

Step-by-step explanation:

Given:

$y = \frac{x² + 3}{ x² - 5}$

To find: dy/dx

Solution:

Quotient formula of Differentiation is used here,which is

$\frac{d}{dx} \bigg( \frac{u}{v} \bigg) = \frac{u \frac{dv}{dx} - v \frac{du}{dx} }{ {v}^{2} }$

So,apply

$\frac{d}{dx} \bigg( \frac{ {x}^{2} + 3}{ {x}^{2} - 5} \bigg) = \frac{( {x}^{2} + 3) \frac{d( {x}^{2} - 5)}{dx} - ( {x}^{2} - 5)\frac{d( {x}^{2} + 3)}{dx} }{ {( {x}^{2} - 5) }^{2} }$

Apply power rule of Differentiation

$\frac{d}{dx} \bigg( \frac{ {x}^{2} + 3}{ {x}^{2} - 5} \bigg) = \frac{( {x}^{2} + 3) (2x) - ( {x}^{2} - 5)(2x)}{ {( {x}^{2} - 5) }^{2} } \\ \\ \frac{d}{dx} \bigg( \frac{ {x}^{2} + 3}{ {x}^{2} - 5} \bigg)= \frac{2x( {x}^{2} + 3 - {x}^{2} + 5) }{ {( {x}^{2} } - 5)^{2} } \\ \\ \because \text{2x \: is \: common \: in \: numerator} \\ \\ \text{cancel \: terms \: to \: simplify} \\ \\ \frac{d}{dx} \bigg( \frac{ {x}^{2} + 3}{ {x}^{2} - 5} \bigg)= \frac{2x \times 8}{ {( {x}^{2} } - 5)^{2} } \\ \\ \boxed{\frac{d}{dx} \bigg( \frac{ {x}^{2} + 3}{ {x}^{2} - 5} \bigg)= \frac{16x}{ {( {x}^{2} } - 5)^{2} }}$

Thus,

Option A is correct.

Hope it helps you.

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