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Fie functia f(x)=1/x(1+lnx). Calculati f'(x)

Răspuns :


[tex]\it f(x) = \dfrac{1}{x}\cdot (1+lnx) \\ \\ \\ f'(x) = \left(\dfrac{1}{x}\right)'(1+lnx) +\dfrac{1}{x} \cdot(1+lnx)' = -\dfrac{1}{x^2}(1+lnx) + \\ \\ \\ + \dfrac{1}{x}\cdot\dfrac{1}{x} = -\dfrac{1}{x^2}(1+lnx) + \dfrac{1}{x^2} = \dfrac{1}{x^2} (-1-lnx+1) = -\dfrac{lnx}{x^2} [/tex]


MindShift
1/x (1+lnx) =>  o simplificam si obtinem  ln(x)+1 totul supra x 
=> aplicam formula de derivare (f/g)' = f'g-fg' totul supra  g patrat
=> [tex] \frac{[ln(x)+1]' (x) - ln(x)+1 [x]'}{x^2} [/tex]
=> [tex]\frac{-ln(x)+( \frac{1}{x} + 0) x-1 }{x^2} [/tex]
=> [tex] -\frac{ln(x)}{x^2} [/tex]

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