Starting with:
#d / dx (-ln(x - (x^2+1)^(1/2)))#
use the chain rule to get:
#d / dx (-ln(x - (x^2+1)^(1/2)))#
#= -1/(x-(x^2+1)^(1/2)) *d/dx (x - (x^2+1)^(1/2))#
use the chain rule once again on the remaining derivative:
#= -1/(x-(x^2+1)^(1/2)) * (1 - 1/2(x^2+1)^(-1/2)(2x))#
Simplify:
#= (x/(x^2+1)^(1/2) -1 )/(x-(x^2+1)^(1/2))#
Note that #1=(x^2+1)^(1/2) / (x^2+1)^(1/2)#, then substitute this for 1:
#= (x/(x^2+1)^(1/2) - (x^2+1)^(1/2) / (x^2+1)^(1/2) )/(x-(x^2+1)^(1/2))#
#= ((x - (x^2+1)^(1/2)) / (x^2+1)^(1/2) )/(x-(x^2+1)^(1/2))#
#= ((x - (x^2+1)^(1/2)) / (x^2+1)^(1/2) ) divide (x-(x^2+1)^(1/2))#
#= ((x - (x^2+1)^(1/2)) / (x^2+1)^(1/2) ) * 1/(x-(x^2+1)^(1/2))#
#= (x - (x^2+1)^(1/2)) / (x-(x^2+1)^(1/2) ) * 1/((x^2+1)^(1/2))#
#= 1 * 1/((x^2+1)^(1/2)) #
#= 1/((x^2+1)^(1/2)) = 1/sqrt((x^2+1)).#
Finally:
#d / dx (-ln(x - (x^2+1)^(1/2))) = 1/sqrt((x^2+1)).#
If you have any questions about the use of the chain rule or any other part of this solution, then please ask.
Rory.
