1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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://en.wikipedia.org/wiki/Partial_derivative
    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

    cnh1995

    User Avatar
    Homework Helper

    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

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

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

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

    And

    ##\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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Partial Differentiation -- y deleted or ignored?
  1. Partial Differentiation (Replies: 11)

  2. Partial differentiation (Replies: 10)

Loading...