Dérivée de $ f(x) = P(x) \pm \frac{k}{x} $
Rappel :
• \( (ax+b)' = a \)
• \( \left(\dfrac{k}{x}\right)' = -\dfrac{k}{x^2} \)
• \( A^2 - B^2 = (A - B)(A + B) \)
Tu dois maîtriser 2 cas:
Exemples:
Si $f(x) = 3x + 7 - \dfrac{2}{x}$, alors $f'(x) = 3 + \dfrac{2}{x^2} = \dfrac{3x^2}{x^2} + \dfrac{2}{x^2} = \dfrac{3x^2+2}{x^2}$.
Si $f(x) = 4x + 7 + \dfrac{25}{x}$, alors $f'(x) = 4 - \dfrac{25}{x^2} = \dfrac{4x^2}{x^2} - \dfrac{25}{x^2} = \dfrac{4x^2-25}{x^2} = \dfrac{(2x-5)(2x+5)}{x^2}$.
Soit la fonction :
\[
f(x) = -7x+2 + \frac{8}{x}
\]