Derivative help please!

• Mathstudent210

  14 Jan 13, 21:54

  Hi!
  
  May I ask you for help?
  What I have is : 

  sqrt(1+x^2)-1 / x

  How to differentiate it with respect to x??

   I've tried to use Combinatorial Calculator here:
  http://www.numberempire.com/derivativecalculator.php?function=sqrt%281%2Bx%5E2%29-1%2Fx&var=x&order=1
  
  And I've got: (sqrt(x^2+1)+x^3)/(x^2*sqrt(x^2+1))
  

   

  Seems like it is the right answer.
  But I want to understand the way of solving it.
  Please show me step by step solution.
  
  Thanks in advance!

  
  

• wee_ws

  15 Jan 13, 08:47

  Hi,

    There are two terms in the expression.

    We can deal with both terms using the fact that

    d/dx [ f(x) ]^n = n [ f(x) ]^(n - 1) . f '(x).

    Essentially, we

    1. bring down the power,

    2. reduce the power by 1,

    3. differentiate the expression inside the power.

    Now d/dx (1 + x^2)^(1/2) = (1/2)(1 + x^2)^(-1/2) . (2x)

                                         = x / sqrt(1 + x^2), and

    d/dx [ x^(-1) ] = -x^(-2).

    Combining, we obtain the answer you have computed previously.

    Thanks.

  Cheers,
  Wen Shih