Question

ln[(x+2)/(x-2)] derivata​

Answer 1

Aici avem trei functii distincte:

a) Functia logaritm

b) x+2

c) x-2

Prima data aplicam regula lantului: f'(g(x)) * g'(x), unde f(x) = functia logaritm natural; g(x) = x+2/x-2 :

$\frac{d}{dx} (ln\frac{x+2}{x-2}) = \frac{1}{\frac{x+2}{x-2}} \frac{d}{dx}(\frac{x+2}{x-2})$

Acum trebuie sa rezolvam g'(x), pe care o rezolvam cu ajutorul regulii catului: [g(x)f'(x) - f(x)g'(x)] / g(x)^2 , unde g(x) = x-2 ; f(x) = x +2

$\frac{(x-2)*1 - (x+2)*1 }{(x-2)^{2}} = \frac{x-2-x-2}{x^2-4x+4}=\frac{-4}{x^2-4x+4}$

In sfarsit, punem totul impreuna:

$\frac{1}{\frac{x+2}{x-2} } * \frac{-4}{(x-2)^2} = \frac{(x-2)*\frac{-4}{(x-2)^2} }{x+2} = \frac{\frac{-4}{x-2} }{x+2}= \frac{-4}{(x-2)(x+2)}=\frac{-4}{x^2-4}$