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Derivative of the square root of xy

  1. Jan 11, 2009 #1
    1. The problem statement, all variables and given/known data

    What is the deriv. of the square root of (xy)?

    2. Relevant equations

    3. The attempt at a solution

    I used the chain rule:

    (1/2)(xy)^(-1/2) times (y + x(dy/dx))

    i am unsure on how to distribute this correctly
  2. jcsd
  3. Jan 11, 2009 #2


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    Science Advisor

    The derivative of [itex](xy)^{1/2}[/itex] with respect to x, and y is a function of x?

    If that is the question, then yes that is correct. I don't know what you mean "how to distribute this correctly". The distributive law is the distributive law: a(b+ c)= ab+ ac.
    Is it the half powers that concern you? [itex](xy)^{1/2}x= (x^{1/2})(x)(y^{1/2}= x^{3/2}y^{1/2}[/itex] and [itex](xy)^{1/2}y= (x^{1/2})(y^{1/2})y= x^{1/2}y^{3/2}[/itex].

    [itex](1/2)(xy)^{1/2}[y+ x dy/dx]= (1/2)x^{1/2}y^{3/2}+ x^{3/2}y^{1/2} dy/dx[/itex]
  4. Jan 11, 2009 #3
    Are you trying to do implicit differentiation? If so treat

    [tex] \sqrt{xy} = \sqrt{x}\sqrt{y} [/tex]

    Then use the product rule, just remember when you differentiate [tex] \sqrt{y} [/tex] to multiply by y'.
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