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General differential question

  1. Oct 2, 2014 #1
    Hey guys. I have a question regarding the differential operator d.

    Say we have an equation d(Z+X*Y^2)=0
    If we want to differentiate the expression in the parenthesis, are these the steps to follow?

    Apply product rule to the second term:

    Here is where I get confused. To simply the 3rd term (X*dY^2), is the simplification this:
    2Y*X*dY or 2*X*dY?
  2. jcsd
  3. Oct 2, 2014 #2


    Staff: Mentor

    Sort of. Assuming that f is a function of x, y, and z, then the total differential df involves the three partials.
    In other words, $$ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz$$
    You should get this:
    $$\frac{\partial (xy^2)}{\partial x} dx + \frac{\partial (xy^2)}{\partial y} dy $$
    For each partial, treat the other variable as if it were a constant. Is that clear?
  4. Oct 2, 2014 #3
    Got it. Thank you.

    I don't know how to write out in the format you did, but for that last term partial(x*y^2)/partialy*dy,
    I was a little confused on how that simplifies.
    We hold x constant for that term, so this gives:
    Does this give
  5. Oct 2, 2014 #4


    Staff: Mentor

    This -- x*2*y*partial(y)/partial_y*dy -- which simplifies to 2xy dy. The partial of y with respect to y is just 1.

    I wrote my previous reply using LaTeX, which isn't too difficult. It looks like this:
    \frac{\partial f}{\partial x}
    Put a pair of $ symbols at front and back, and it renders like this:
    $$\frac{\partial f}{\partial x}$$
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