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Differentiating exponentials

  1. Jan 29, 2005 #1
    Hey again,

    well i just studying several vaiable calculus, and encountered the problem of finding the gradient of the scalar field:

    f = ye^(xy)

    now I could successfully find the i component (y^2.e^(yx))
    but im having some trouble with the j component.

    [tex]f = ye^{(yx)}[/tex]

    if my understanding is correct to differentiate this wrt to y, we treat everything else (x) as if it were a constant.
    now when i encouner stuff like this, i plug in an arbitrary number for x, such as 2, and continute like that

    [tex]f = ye^{(2y)}[/tex]

    now my intuition says [tex]df/dy = 2ye^{(2y)}[/tex]
    and plugging x back in for the 2:
    [tex]df/dy = xye^{(xy)}[/tex]

    but this is obviously incorect, with both the solutions and a calculator giving the answer of:
    (y.x + 1)e^(xy) or yxe^(xy) + e^(xy)

    i have no doubts its correct but what is the procedure to get the additional exponential term? and under what conditions is it +2, +3, etc?

    i tried searching the internet but just found examples wthout the leading variable (in this case y in front of the e).
    and another thing, is my approach of answering partial DE's okay? (e finding something wrt y, replacing the x's and z's with integers, then diff'ing?
    i guess i find it difficult just looking at something like [tex]f = ye^{(xy)}[/tex] and instantly finding df/dx and df/dy. are there ay other approachs out there?

    thanks for reading
  2. jcsd
  3. Jan 29, 2005 #2


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    Homework Helper

    It really doesn't matter how u denote (this in the case in which u denote) the variable(s) kept constant in the partial differentiation,the important thing is to apply the rules correctly.
    Unfortunately u didn't...
    Compute the derivative of the product
    [tex] f(x)g(x) [/tex]

    wrt to "x".Then use this rule to CORRECTLY differentiate your formula...

  4. Jan 30, 2005 #3
    ahh that's right, good old product rule.
    deadset it's been at least 3 years since ive needed to use it lol

    thanks alot Daniel
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