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 40,784
    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 3,378
    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 3,378
    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 a link to this question via email, Google+, Twitter, or Facebook

Have something to add?