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Second-Order Partial Derivative of a Parametric Function

  1. Feb 7, 2012 #1
    The problem is from an online homework assignment. I know it's probably fairly simple, but my brain isn't grasping it right now for some reason.


    [The Problem]

    We know:

    r(t) = <3t2 - 8t + 3, -9t2 + 2t + 7>

    And we are asked to find d2y/dx2.


    [Background Information]

    My understanding of d2y/dx2 is that it is the second derivative with respect to x (the first derivative of the function having been with respect to both x and y).

    In other words, it's broken down like this:

    (d/dx)(dy/dx) = d2y/dx2
    The derivative (with respect to x) of the first derivative of the parametric function, r(t), is equal to that mess on the right hand side of the equation.

    [Attempt at a Solution]

    So we know that:

    (dy/dx) = (dy/dt) / (dx/dt) (From the textbook.)

    And the above function, r(t), can be broken down into two parts:

    x(t) = -9t2 + 2t + 7
    y(t) = 3t2 - 8t + 3

    Therefore:

    (dx/dt) = 6t - 8
    (dy/dt) = -18t + 2

    And

    (dy/dx) = (-18t + 2) / (6t - 8)


    So, now, here's where I feel like I'm guessing a little bit. Following that logic, would d2y/dx2 = (6) * [(-18t + 2)/(6t - 8)]? :rolleyes:



    Heh, this is probably a silly question, but thanks very much in advance for any help! :smile:
     
  2. jcsd
  3. Feb 8, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

    dy/dx is correct, but I can not follow your logic afterwards. d2y/dx2 =d(dy/dx)/dx = (d(dy/dx)/dt)/(dx/dt). Are you sure you did the derivative of the fraction correctly?



    ehild
     
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