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Partial Differentiation Question

  1. Mar 5, 2009 #1
    I have a basic question about taking partial derivatives.

    Say I have a function of 3 variables and i want the derivative of only one. Do I take the derivative of the one variable and HOLD THE OTHER TWO CONSTANT? Or, do I take the derivative of the variable and TREAT THE OTHER TWO AS CONSTANTS?

    For example

    If I have f(x,y,z) = xy2+z3 and I want to find fx

    Does that mean I get 2xy or 2xy+z3?

    Like does that z drop out or do I literally not touch it? This isn't a homework question. I'm just trying to understand what the rule is.
     
  2. jcsd
  3. Mar 5, 2009 #2
    well, what is the definition of a partial derivative of a function of several variables? What is the rule of taking the derivative with respect to a specific variaable?

    2xy is your answer. Although i initially didn't know the difference between:"HOLD THE OTHER TWO CONSTANT" vs "TREAT THE OTHER TWO AS CONSTANTS", i believe, the second one applies.

    You have to treat the other variables as constants.

    A side note: if we have a function of two variables, say, z=f(x,y), and we want to take the partial derivative with respect to x, fx, then geometrically what it means is that we are finding the slope of the tangent lines at any point x on the traces Ci in the vertical planes y=yo parallel to the xz plane. What this means is tha on background we are treating y as a constant.
     
  4. Mar 5, 2009 #3
    Huh? Isn't the answer

    [tex]y^{2}[/tex]
     
  5. Mar 5, 2009 #4
    Well, if you are taking the partial with respec to x, then yes. But if you are taking the partial with respec to y, then it is 2xy.
     
  6. Mar 5, 2009 #5
    Sorry, I guess I have the notation confused, I thought the fx was the derivative wrt x.
     
  7. Mar 5, 2009 #6
    No,you are correct [tex]f_x[/tex] is the partial with respect to x.
     
  8. Mar 6, 2009 #7
    Um, so the answer to the original question is neither. You get y2.
     
  9. Mar 6, 2009 #8
    OOO, i just saw that he was differentiating wrt to x. Yep, that is the correct answer.
     
  10. Mar 6, 2009 #9

    HallsofIvy

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    What do you understand as the difference between "HOLD THE OTHER TWO CONSTANT" and "TREAT THE OTHER TWO AS CONSTANTS"?

    Neither as you have been told. Whether you "TREAT THE OTHER TWO AS CONSTANTS" or
    "HOLD THE OTHER TWO CONSTANT", what is the derivative of f(x)= xb2+ c, where b and c are constants.
     
  11. Mar 10, 2009 #10
    Yeah I made a mistake, sorry about that.
     
  12. Mar 10, 2009 #11
    f'(x) = 2xb

    Yeah I geometrically understand what it is to find the partial derivative. Basically you're intersecting a plane with a surface and finding the derivative of the curve that the intersection of the plane and the surface makes.

    But I was mistaken earlier.

    Like If I had

    f(x,y,z) = x3yz+3yz-2y+z and I wanted fx Do I get

    fx(x,y,z) = 3x2yz
    OR
    fx(x,y,z) = 3x2yz +3yz-2y+z

    I guess the first one is correct because since the rest are treated as constants, they simply drop out.

    Also, how would I do something like say:

    f(x,y,z) = e(x2-y2+z2), find fx

    I'm just trying to understand the rules of partial differentiation. Thanks for your help so far!
     
  13. Mar 10, 2009 #12
    Same thing as above, just that in this case you need to apply chain rule! THat is, you still are considering y,z as constants, but the extra twist here is that you have a function,namely e, raised to a function(which is not simply an independent variable), so as i mentioned, you'll need chain rule here.
     
  14. Mar 11, 2009 #13

    HallsofIvy

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    What is the derivative of
    [tex] f(x)= e^{x^2+ a}[/tex]?
     
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