Derivatives of square roots

  1. How can i find the derivative of a function like this:
    f(x) = sqrt( 1 - x² )
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    Do you know how to write a square root with exponents?
     
  4. (1 - x² )^(1/2) ?
     
  5. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    That's right! Now, you just need to apply what you know about differentiating expressions like that.
     
  6. si will that be:
    1/2((1-x²)/2)^(-1/2)

    ?
     
  7. HallsofIvy

    HallsofIvy 40,967
    Staff Emeritus
    Science Advisor

    Not quite. You have one too many "1/2"s (you don't want that "/2" inside the square root and you didn't use the chain rule.

    You need to multiply by the derivative of 1-x2.
     
    Last edited: Jul 17, 2009
  8. Hi,

    I have a similar problem, I need to differentiate sqrt(x^2 + y^2) in terms of x and y.
    Starting this I took the simple step (x^2 + y^2)^(1/2)...

    My next step is a guess and I am lost after it....(1/2)(x^2 + y^2)(-1/2)....

    Any help would be much appreciated.
     
  9. jbunniii

    jbunniii 3,379
    Science Advisor
    Homework Helper
    Gold Member

    If by "in terms of x and y", you mean you want to calculate the partial derivatives, then for the partial derivative with respect to x, treat y as a constant and differentiate with respect to x as you normally would a function of one variable. For the partial derivative with respect to y, treat x as constant.
     
  10. thanks, would I do this by...

    df/dx = 1/2(x^2 + y^2)^(-1/2).2x = x/sqrt(x^2 + y^2) and...

    df/dy = 1/2(x^2 + y^2)^(-1/2).2y = y/sqrt(x^2 + y^2) ?

    thanks for your help so quickly.
     
  11. jbunniii

    jbunniii 3,379
    Science Advisor
    Homework Helper
    Gold Member

    Looks good to me.
     
  12. What if i have a problem similar to these however now its 4/ ^5sqrt(x^5)
     
  13. If that is meant to be 4^(5(sqrt(x^5))), then you can easily rewrite this to equal
    4^(5(x^(5/2)) And using what you know from differentiating exponentials and chain rule, you should be able to get the rest.
     
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?