L'énoncé
Question 1
Soit \(f(x) = (ln(x))^2-6ln(x)\), pour $x>0$
Déterminer la dérivée de $f$.
\(f'(x) = 2\times \dfrac{1}{x} \times \ln(x)-6\times\dfrac{1}{x}\)
\(f'(x) = \dfrac{2\ln(x)-6}{x}\)
Question 2
\(f(x) = 2x+\ln\left(\dfrac{x+1}{x}\right)\), pour $x>0$
\(f(x) = 2x+\ln\left(\dfrac{x+1}{x}\right)\)
\(f(x) = 2x+\ln(x+1)-\ln(x)\)
Ainsi :
\(f'(x) = 2+\dfrac{1}{x+1}-\dfrac{1}{x}\)
\(f'(x) = \dfrac{2x(x+1)+x-x-1}{x(x+1)}\)
\(f'(x) = \dfrac{2x^2+2x-1}{x(x+1)}\)
\((\ln(u))' = \dfrac{u'}{u}\)
Question 3
\(f(x) = \dfrac{x+\ln(x)}{x^2}\)
\(f(x) = \dfrac{x+\ln(x)}{x^2} = \dfrac{1}{x}+\dfrac{\ln(x)}{x^2}\)
Ainsi: \(f'(x) = -\dfrac{1}{x^2}+\dfrac{\frac{1}{x} \times x^2-\ln(x) \times 2x}{x^4}\)
\(f'(x) = -\dfrac{1}{x^2}+\dfrac{x-2x\ln(x)}{x^4}\)
\(f'(x) = -\dfrac{1}{x^2}+\dfrac{1-2\ln(x)}{x^3}\)
\(f'(x) = \dfrac{1-2\ln(x)-x}{x^3} \)