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Finding a function for rate of change

  1. Jul 12, 2013 #1
    Hello!

    I was wondering if somebody could help me develop an answer to the question below.

    I would like to calculate a rate of change of the following function:

    θ = arctan (a * (tan β)) where "a" is a constant.

    By the way, the shape of the curve -- angle alpha as a function of angle Beta --resembles an arm of a parabola. Angle Beta increases with angle alpha, but less so with increasing values.

    I would like to derive a function that would describe the instantaneous rate of change at each point of that curve. It would be great if the function would say: hey, this is an arm of a parabola! :)

    Unfortunately I took math a very long time ago... I believe I would
    need to find the derivative of the function? Could you kindly help me out with this?
    thnx

    Penny
     
  2. jcsd
  3. Jul 12, 2013 #2

    HallsofIvy

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    The whole point of the "derivative" is that it is the rate of change of a function. The derivative of arctan(x), with respect to x, is [itex]1/(1+ x^2)[/itex]. The derivative of tan(x) is [itex]sec^2(x)[/itex].

    So, by the "chain rule", with f= arctan(u) ,u= a tan(x), [itex]df/du= 1/(1+ u^2)[/itex] and [itex]du/dx= a sec^2(x)[/itex], so [itex]df/dx= (df/du)(du/dx)= (1/(1+ u^2))(a sec^(x)= (a sec^2(x))/(1+a^2tan^2(x)[/itex].
     
  4. Jul 13, 2013 #3
    ^ What HallsOfIvy said

    If you're looking for an easy way to find derivatives and do other mathsy things, look up wolfram alpha. It's a website that is quite excellent.
     
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