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Question regarding a derivative

  1. Sep 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Write the derivative of y = (x2+4x+3)/(x1/2)

    I got the correct answer, but my question is, why can't I rewrite this as:

    y = (x^2+4x+3)*(1/x1/2)

    Then see my attempted solution for the result...

    2. Relevant equations
    y = (x2+4x+3)/(x1/2)

    3. The attempt at a solution

    y = (x^2+4x+3)*(x-1/2)
    d/dx = (2x+4)*(-1/2x-3/2)


    The problem is that this seems to drop out a term from the equation, because the answer that I got when treating each term as being divided by x1/2 gave me:

    d/dx = 1.5x.5+2x-.5-3/(2x(x.5)), which was the answer in the book. Yeesh, sorry this is difficult to read over forums. I tried to use decimals to make it more readable over the forums, but I don't know if it helped. Maybe next time I'll post an image of my work. Sorry.

    Anyways, when I did the method I show above, I get a different equation entirely because the term drops out due to the constant becoming a 0 in the derivative. However, it seems that I've followed all the laws correctly that I've been taught. That is, why can't I take each term as a derivative, and multiply it by another derivative?
     
  2. jcsd
  3. Sep 25, 2015 #2
    Because then you need to use the product rule. If you do that, you'll get the right answer.
     
  4. Sep 25, 2015 #3
    Okay, thanks. I just realized that I am going to come across that in the next section. :D
     
  5. Sep 25, 2015 #4

    Mark44

    Staff: Mentor

    Two comments:
    1. In your second equation what you have is the product of the derivatives of the factors in the first equation. That's not how the product rule works. IOW, ##\frac d {dx} f(x) \cdot g(x) \ne f'(x) \cdot g'(x)##
    2. In your second equation you have "d/dx = ..." This is incorrect. The symbol ##\frac d {dx}## is an operator that requires something to operate on, and that appears just to the right of this operator. For the second equation, "dy/dx = " would be appropriate, but "d/dx = " is not, since you haven't shown what it is that you're taking the derivative of. It's a bit like writing √ = 3 or sin = 0.5.
     
  6. Sep 25, 2015 #5

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Please, please do NOT post an image; a typed version is much better, and is the PF standard.

    Anyway, it might be easiest of all if you re-write your function before doing the derivative:
    [tex] y = \frac{x^2 + 4x + 3}{x^{1/2}} = \frac{x^2}{x^{1/2}} + \frac{4x}{x^{1/2}} + \frac{3}{x^{1/2}} \\
    \\
    = x^{3/2} + 4 x^{1/2} + 3 x^{-1/2} [/tex]
    In this last form you can just differentiate term-by-term, and so do not need the product or quotient rules.
     
  7. Sep 25, 2015 #6

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    The derivative should be denoted by dy/dx, not d/dx. dy/dx means f'(x), where f is the function defined implicitly by the previously stated relationship between x and y.
     
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