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Homework Help: Derivatives of integrals and inverse functions

  1. Sep 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the derivative of:

    [itex]1. f(x)=arccos(5x^3)[/itex]

    [itex]2. f(x)=∫cos(5x)sin(5t)dt[/itex] when the integral is from 0 to x

    2. Relevant equations

    Chain rule, dy/dx=dy/du*du/dx

    3. The attempt at a solution

    For the first one, I can just take 5x^3 as u and then apply the chain rule to get


    but maple TA keeps telling me that it's incorrect...

    As for the second one, I'm not sure how to approach something like that when x is actually in the integral, as in the cos(5x). I mean, when I'm applying the chain rule...do I have to include cos in my choice of u?
  2. jcsd
  3. Sep 15, 2012 #2
    Check the derivative of arccos again.

    As for the second one, when you have something like this:
    [tex]\int_{g(x)}^{h(x)} f(t)dt[/tex]
    It's derivative is equal to
    [tex]f(g(x)) \cdot g'(x)-f(h(x)) \cdot h'(x)[/tex]
  4. Sep 15, 2012 #3
    Ah, so the derivative of the first one should be

  5. Sep 15, 2012 #4
    Yep! :smile:
  6. Sep 16, 2012 #5
    Thanks. But for the second one, does it mean that I can just treat the cos(5x) as a constant? i.e. I only have to multiply by the derivative of x?

    I know what to do if I have something like

    [itex]d/dx∫cos(5t)sin(5t)dt[/itex], where the integral is from 0 to x. I just differentiate by basically putting the x into the ts to get


    But when the cos5t in the integral becomes cos5x, do I just do this?

  7. Sep 16, 2012 #6
    Yep, just take out the cos(5x) from the integral.
    You are left with
    [tex]f(x)=cos(5x) \int_{0}^{x} sin(5t)dt[/tex]
    You can either differentiate by the chain rule or better, solve the integral, put the limits and differentiate.
  8. Sep 16, 2012 #7
    Great, I finally got it, thanks again :)
  9. Sep 16, 2012 #8


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    Don't forget, [itex]\displaystyle \int_{0}^{x} \sin(5t)dt[/itex] is a function of x, so you have to take that into account when you differentiate [itex]\displaystyle \cos(5x)\int_{0}^{x} \sin(5t)dt\ .[/itex]
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