Posted by Heather on Thursday, October 4, 2012 at 1:10pm.
Find the derivative of ln(x+(x^21)^(1/2)).

Calculus 2  Reiny, Thursday, October 4, 2012 at 1:53pm
let y = ln [ x + (x^2  1)^(1/2) ]
dy/dx = 1/(x + (x^2  1)^(1/2) * (1 + (1/2)(x^2  1)^(1/2) (2x) )
= 1/(x + √(x^2  10 ) * (1 + x/√(x^2  1) )
= 1/(x + √(x^2  10 ) * (√(x^2 1) + x)/√(x^21)
let's multiply top and bottom by x  √(x^2  1) , thus rationalizing the denominator in the first part
= 1/(x + √(x^2  10 ) * (√(x^2 1) + x)/√(x^21) * [x  √(x^2  1)]/[x  √(x^2  1)]
= (x^2  x^2 + 1)/( (√x^2  1)(x^2  x^2 + 1) )
= 1/√(x^2  1)
Whewww!

Calculus 2  Steve, Thursday, October 4, 2012 at 2:10pm
you can make things a little less complicated if you recognize that
ln(x+√(x^21)) = arccosh(x)
d/dx arccosh(x) = 1/√(x^21)
Reiny's excellent work shows how it's done.
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