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Derivative of sec(xy)

  1. Mar 12, 2012 #1
    1. The problem statement, all variables and given/known data


    I'm getting stuck on this problem. Not only do I not know how to proceed, I don't understand why I need to put it into that final form.


    trigimplicit1.png



    EDIT: Okay, i get why it needs to be in that final form...obviously because dy/dx needs to be isolated. However, I still don't know how to proceed.
     
    Last edited: Mar 12, 2012
  2. jcsd
  3. Mar 12, 2012 #2
    Solving this problem involves using the product rule and the chain rule, as well as trigonometric derivatives. Have you learned these yet?

    Hint: derivative of sec(x) = sec(x)tan(x)
     
  4. Mar 12, 2012 #3

    I already derived sec(x). That is not where I'm confused. The part where I'm getting stuck is after the function is already derived and I have to get it into the final form ...after the arrow in the image.
     
  5. Mar 12, 2012 #4

    SammyS

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    Staff Emeritus
    Science Advisor
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    Gold Member

    There are several algebra steps involved.

    Distribute the sec(xy)∙tan(xy) .

    Get all terms with dy/dx on the left hand side. Factor out dy/dx .

    ...
     
  6. Mar 12, 2012 #5
    What I'm trying to say is that the way you derived it is probably making it more difficult. Use sec(xy)tan(xy) and use the chain rule for the (xy). Then follow what SammyS said.
     
  7. Mar 12, 2012 #6

    Mark44

    Staff: Mentor

    LearninDaMath,
    You are making the same mistake you made on a thread of few days ago. The first line in your post says
    "y = sec(xy)"
    This makes no sense because the right side also involves y.

    Your work should look something like this:

    d/dx(sec(xy)) = sec(xy)*tan(xy) *d/dx(xy)
    = ...

    For the derivative at the end of the first equation above, you need to use
    1) the product rule, followed by
    2) the chain rule

    Tip: make life a little easier on yourself by using the fact that d/dx(sec(x)) = sec(x)*tan(x). You are making the problem more difficult than it needs to be by not using this formula.
     
  8. Mar 12, 2012 #7
    tal444, thanks for your help. Sorry that my question may not have been specific enough. Sometimes it feels more efficient for me to scan my work directly to the computer instead of trying to figure out the latex symbols. However, I am getting better at it though.

    SammyS, thanks. I got it now. Appreciate your feedback as always.
     
  9. Mar 12, 2012 #8

    Mark44

    Staff: Mentor

    But it's not more efficient for us, since we can't insert a comment in at exactly the right spot.
     
  10. Mar 12, 2012 #9
    No problem, glad I could help.
     
  11. Mar 12, 2012 #10

    Mark, I was typing my comment during the time you posted yours. Just saw that your post now, thanks for providing information on this. The question of my post comes directly from the issued text book. The question is like this:

    Find dy/dx by implicit differentiation.

    sec(xy) = y

    and that is exactly as its written.

    I recall your feedback from that previous posted saying that I can not refer to sec(xy) = y as: f(x) = sec(xy)

    and this is because sec(xy) is not a function. However, it is still an equation. And implicit differentiation can be applied to both sides of an equation, even if that equation involves the same variable on both sides.

    I believe this was what I took away from the recent help you provided on that previous thread.
     
  12. Mar 12, 2012 #11
    Oh, I didn't see it from the view of those who are providing the instruction/assistance, good point. I'll start transitioning over to latex format.
     
  13. Mar 12, 2012 #12

    Mark44

    Staff: Mentor

    And this is why we ask that you use the template, with the first part being the problem statement. In essence, the problem is asking you to implicitly differentiate the equation y = sec(xy). The equation is not defining y as a function of x and itself, which is what I thought you were trying to do.
    But it is a function - a function of two variables.
    Yes, good.
     
  14. Mar 12, 2012 #13

    Mark44

    Staff: Mentor

    Thank you. I appreciate it, and I think others will as well.

    LaTeX isn't very hard, and if all you need to do are exponents, you can do them without LaTeX. Just click the Go Advanced button below the input area, which opens an advanced menu along the top of the input area. The X2 can be used for exponents, and there are a bunch of useful symbols along the right side, including ∫, Ʃ, ∞, √, and several others.
     
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