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Purpose of the derivative of the inverse function

  1. Nov 10, 2018 #1
    1. The problem statement, all variables and given/known data
    In calculus, I learn that the derivative of the inverse function is

    g'(x) = 1/ f'(g(x))

    2. Relevant equations
    So..

    3. The attempt at a solution

    Can someone give me an example of where I need to know this, or is this just a math exercise. Is there a relatively simple physics example?
     
  2. jcsd
  3. Nov 10, 2018 #2

    FactChecker

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    Sure. Suppose you are controlling an engine speed with a dial. If you want to increase the speed by 3 rpm, how much should you increase the dial setting?

    PS. Keep in mind that this gives a linear estimate at that setting. It maybe very different at another setting.
     
  4. Nov 10, 2018 #3
    Don't go away for long. I will be back in4 hours.
     
  5. Nov 10, 2018 #4

    FactChecker

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    This property of the derivative of the inverse function is just a simple observation of what a slope is when looked at from another direction. As such, it has applications everywhere.
     
    Last edited: Nov 10, 2018
  6. Nov 10, 2018 #5

    Math_QED

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    What is the derivative of ##\arccos x##? Good luck with using the definition.
     
  7. Nov 10, 2018 #6
    OK, finding the derivative of arccos(x)
    given... f(x) = cos(X) and g(x) = cos^-1(x)

    then g'(x) = 1/f'(g(x))

    g'(x) = 1/-sin(g(x))

    g'(x) = 1/-sin(cos^-1(x))

    then after some trig substitutions we get this is equal to ##-1/\sqrt(1-u^2)##or something like this.
     
  8. Nov 10, 2018 #7

    Math_QED

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    This is one of the numerous examples where it is useful.
     
  9. Nov 10, 2018 #8
    Could someone give me a few more examples where finding the derivative of the inverse function is useful.
     
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