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Partial Differentiation -- y deleted or ignored?

  1. May 2, 2015 #1
    1. The problem statement, all variables and given/known data
    Given: z = f(x,y) = x^2-y^2
    To take the partial derivative of f with respect to x hold y constant then take the derivative of x.
    ∂f/∂x = 2x

    What I don't understand is why such would equal 2x, when the y is still there it just isn't variable and is ignored. Wouldn't it be:

    2x - y^2

    where y is a constant squared?
    In another example this kind of thing occurred:
    z = f(x,y) = x^3*y
    ∂f/∂x = 3x^2*y

    where y is a constant unknown.

    So why the different pertaining to y in these two answers?

    Thanks so much, Bertrand Russell
    2. Relevant equations
    None really.

    3. The attempt at a solution
    http://www.centerofmath.org/mc_pdf/sec2_1.pdf [Broken]
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. May 2, 2015 #2


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    We take the partial derivative of the entire function..So for f(x,y)=x2-y2, the derivative of x^2 is 2x but the derivative of y^2 w.r.t.x must be 0 since it is constant..Same goes for 3x^2*y..
    Last edited by a moderator: May 7, 2017
  4. May 2, 2015 #3
    Got it. Thanks so much.
  5. May 2, 2015 #4


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    First, in both cases y is treated as a constant. For example:

    ##\frac{d}{dx}(x^2 + a^2) = 2x##


    ##\frac{d}{dx}(x^3a) = 3xa##

    On your main point: a partial derivative (wrt x) is the gradient of the curve you get when you hold y constant. You could think of taking y = 0, 1, 2... and see what curves you get. Whatever value of y you choose, you get a parabola ##z = x^2 + n^2## where ##n## is whatever value of y you are considering. This curve has the usual derivative and does not get steeper as y increases: it's the same shape for any value of y.

    Whether y appears in the partial derivative depends on whether the value of y affects the derivative. For ##z = x^2 + y^2## it doesn't: the partial derivative is independent of y. But for ##z = x^3y## the partial derivative wrt x does depend on y.

    It's exactly the same as whether a constant ##a## affects the ordinary derivative of a function of x.
    Last edited by a moderator: May 7, 2017
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