Question

the derivative of log[e^x{x-2/x+2}^3/4]

please explain the solution step by step​


fasttt i want the answer now I have to do it

Answer 1

Step-by-step explanation:

Given,

$y = log( {e}^{x}( \frac{x - 2}{x + 2} )^{ \frac{3}{4} } )$

Using various properties of log,we get,

$y = log( {e}^{x} ) + log( ({ \frac{x - 2}{x + 2} })^{ \frac{3}{4} } )$

$= > y = x log(e) + \frac{3}{4} (log(x - 2) - log(x + 2) )$

$= > y = x + \frac{3}{4} log(x - 2) - \frac{3}{4} log(x + 2)$

Differentiating both sides, we get,

$= > \frac{dy}{dx} = 1 + \frac{3}{4(x - 2)} - \frac{3}{4(x + 2)}$

$= > \frac{dy}{dx} = 1 + \frac{3}{4} ( \frac{(x + 2) - (x - 2)}{(x + 2)(x - 2)} )$

$= > \frac{dy}{dx} = 1 + \frac{3}{4} . \frac{4}{ ({x}^{2} - 4) }$

$= > \frac{dy}{dx} = 1 + \frac{3}{( {x}^{2} - 4) }$

$= > \frac{dy}{dx} = \frac{ {x}^{2} - 1 }{ {x}^{2} - 4 }$