oblicz pochodną f'(x)=1-lnx/1+lnx
1 - lnx
f(x) = -------------
1 + lnx
(1 - lnx) ` * (1 +lnx) - (1 - lnx) * (1 + lnx) ` (0 - 1/x) * (1 + lnx) - (1 - lnx) * (0 + 1/x)
f`(x) = ---------------------------------------------------- = -------------------------------------------------------- =
(1 + lnx)² (1 + lnx)²
-1/x( lnx + 1) - (1 - lnx) * 1/x -1/x lnx - 1/x - 1/x + 1/x lnx -2/x -2
------------------------------------- = ------------------------------------- = ------------------ = ---------------
(1 + lnx)² (1 + lnx)² (1 + lnx)² x(1 + lnx)²
(lnx)'= 1/x
f '(x)=(1-lnx)' (1+lnx) - (1-lnx)(1+lnx)'
(1+lnx)²
= (0 -1/x)(1+lnx) - (1 - lnx)( 0+ 1/x)
-1 - lnx - 1 + lnx
= x x x x =
= -2
x(1+lnx)²
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1 - lnx
f(x) = -------------
1 + lnx
(1 - lnx) ` * (1 +lnx) - (1 - lnx) * (1 + lnx) ` (0 - 1/x) * (1 + lnx) - (1 - lnx) * (0 + 1/x)
f`(x) = ---------------------------------------------------- = -------------------------------------------------------- =
(1 + lnx)² (1 + lnx)²
-1/x( lnx + 1) - (1 - lnx) * 1/x -1/x lnx - 1/x - 1/x + 1/x lnx -2/x -2
------------------------------------- = ------------------------------------- = ------------------ = ---------------
(1 + lnx)² (1 + lnx)² (1 + lnx)² x(1 + lnx)²
(lnx)'= 1/x
f '(x)=(1-lnx)' (1+lnx) - (1-lnx)(1+lnx)'
(1+lnx)²
= (0 -1/x)(1+lnx) - (1 - lnx)( 0+ 1/x)
(1+lnx)²
-1 - lnx - 1 + lnx
= x x x x =
(1+lnx)²
= -2
x(1+lnx)²