Sec 5.8 Page 386 # 53
y = (1/2)*[(1/2)Ln
((x+1)/(x1)) +
arctan x]



Things needed
to do the problem.
(d/dx)[arctan u] =
[u' / (1
+ u^{^2})]
Let u = x,
then u ' = 1
therefore:
(d/dx)[arctan u] =
[1 / (1 + x^{^2})]
More things
needed to do the problem.
(d/dx)[Ln x] =
(1/x)


Simplify
the problem, use rules about Natural Logs
Also
factor out the constant in front of the Ln..

y = (1/2)*(1/2)*[(Ln(x+1)
 Ln(x1)] + [(1/2) arctan x]



Simplify
the problem by combining (1/2)*(1/2). 
y = (1/4)*[(Ln(x+1)
 Ln(x1)] + [(1/2) arctan x] 


Now
diff of eqn:

y
' = (1/4)*[1/(x+1)
 1/(x1)]
+ [(1/2) * (1 /
(1 + x^{^2}))]



Get
a common denominator so terms can be combined 

(x^{^2}
+ 1) 

Common
denominator 1st part
[(1/(x+1))
* (x1)] = [(x1) / (x^{^2}1)]


Common
denominator 2nd part
[(1/(x1))
* (x+1)] = [(x+1) / (x^{^2}1)]


Set
back into equation.

y
' = (1/4)*[((x1)/(x^{^2}1))
 ((x+1)/(x^{^2}1))] + [(1/2)*(1
/ (x^{^2}
+ 1))]

Then
combine and simplify
(1/4)*[2/(x^{^2}1))]
+ [(1/2)*(1 / (x^{^2}
+ 1))] 
Simplify
the problem more.
(1/2)*[(1/(x^{^2}+1))]
+ [(1/2)*(1 / (x^{^2}
+ 1))] 

Factor
out (1/2) to simplify 
(1/2)*[(1/(x^{^2}1))
+ (1 / (x^{^2}
+ 1))] 


Get
a common denominator so terms can be combined 

(x^{^4}
 1)


Get
a common denominator of 1st part
(1/(x^{^2}1)*[
(x^{^2}+1)/(x^{^2}+1)]
= (x^{^2}+1)/
(x^{^4}
 1)


Get
a common denominator of 2nd part
(1 / (x^{^2}
+ 1))*[ (x^{^2}1)/(x^{^2}1)]
= (x^{^2}1)/(x^{^4}
 1) 
(1/2)*[((x^{^2}+1)/
(x^{^4}
 1)) + ((x^{^2}1)/(x^{^4}
 1))]
(1/2)*[((x^{^2}+1)*(x^{^2}1))/
(x^{^4}
 1)]
(1/2)*[(2)/ (x^{^4}
 1)]


Final
solution:
y '
(x) = [1/(x^{^4}
 1)] 
