Question

How do you differentiate `f(x)=(1-lnx)/(1+lnx)` using the quotient rule?

Answer 1

`d/dx (1-lnx)/(1+lnx) = - ( 2 ) / ( x(1+lnx)^2 )`

Explanation:

Using the quotient rule, we have:

`d/dx (1-lnx)/(1+lnx) = ( (1+lnx)(d/dx (1-lnx)) - (1-lnx)(d/dx(1+lnx)) ) / (1+lnx)^2`

`" " = ( (1+lnx)(-1/x) - (1-lnx)(1/x) ) / (1+lnx)^2`

`" " = ( (-1/x)( 1+lnx + 1-lnx) ) / (1+lnx)^2`

`" " = ( (-1/x)( 2) ) / (1+lnx)^2`

`" " = - ( 2 ) / ( x(1+lnx)^2 )`