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Weird derivative conflict

  1. Sep 7, 2004 #1
    I am having a conflict with two different ways of finding a derivative.
    Here is the function:
    y=10*sinpi(.01x-2.00t)

    Yes, that pi is after sin, but not in the paranthesis. This is how the prof gave it to us. This may be my problem, how I am treating the pi. I figure it was factored out of the parenthesis. So, to find the partial derivative WRT t by hand I do this:

    y=10*sin(.01pi*x - 2.00pi*t) I multiplied the pi into the ()
    dy/dt = -2.00pi*10*cos(.01pi*x - 2.00pi*t) used the chain rule
    dy/dt = -20pi*cos(.01pi*x - 2.00pi*t) final result

    That is my result. I check this in Matlab by entering the following:

    >> syms x t
    >> diff(10*sin(pi*.01*x-pi*2*t),t)
    ans =
    -20*cos(1/100*pi*x-2*pi*t)*pi

    So with that I am happy. Now the tricky question.
    If I enter this same thing to my TI-89, I get:
    -62.8319*cos(2pi*t - .031416*x)

    Now...it just hit me that you can transpose the items inside the paranthesis of cosine, and it is the same result. Ok, duh. I don't want to delete everything I just typed. My next question...

    Am I treating the pi correctly to begin with? Is it correct to multiply it into the () like that? If not, what should I do with it? Is there an easier way?
    Thanks.
     
  2. jcsd
  3. Sep 7, 2004 #2

    Tide

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    You can certainly multiply it out - it's just the distributive property of numbers.
     
  4. Sep 8, 2004 #3

    HallsofIvy

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    However, it is simpler just to treat [itex]\pi[/itex] as a multiplier:

    The derivative of sin x is cos x so the derivative of [itex] sin(\pi x)[/itex] is [itex]cos(\pi x)[/itex] times the derivative of [itex]\pi x[/itex] (that's the "chain" rule) which is just [itex]\pi[/itex]: the derivative of [itex] sin(\pi x)[/itex] is [itex]\pi cos(\pi x)[/itex].

    The partial derivative of [itex]sin(\pi(0.01x- 2.00t))[/itex], with respect to t, is just that times the derivative of 0.01x- 2.00t with respect to t: [itex]-2.00\pi cos(\pi (0.01x- 2.00t)[/itex].

    Finally, the derivative of [itex]sin(\pi(0.01x- 2.00t))[/itex] is, of course, just 10 times that: [itex]-20.00\pi cos(\pi(0.01x-2.00t)[/itex].
     
  5. Sep 8, 2004 #4
    Thanks

    Thank you both for responding.
    I see what Halls has done, and yes, I think that is simpler. It just didn't occur to me to put pi into the () one time, instead of multiplying to each term.
    Thanks.
     
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