Differentiation with a bit of trig?

Discussion in 'Differentiation and Integration' started by akw, Sep 12, 2022.

  1. akw

    akw

    Joined:
    Apr 9, 2022
    Messages:
    4
    Likes Received:
    0
    Hi,

    Please see the attached image.

    The step that I don't understand is marked. Please could someone explain how this magic happens?
     

    Attached Files:

    akw, Sep 12, 2022
    #1
    1. Advertisements

  2. akw

    MathLover1

    Joined:
    Jun 27, 2021
    Messages:
    2,989
    Likes Received:
    2,883
    upload_2022-9-20_16-56-41.gif

    upload_2022-9-20_16-57-2.gif

    upload_2022-9-20_16-58-3.gif
    upload_2022-9-20_16-58-32.gif
     
    MathLover1, Sep 20, 2022
    #2
    1. Advertisements

  3. akw

    akw

    Joined:
    Apr 9, 2022
    Messages:
    4
    Likes Received:
    0
    Thanks :) But I meant the previous step. I should have been clearer.
     
    akw, Sep 21, 2022
    #3
  4. akw

    MathLover1

    Joined:
    Jun 27, 2021
    Messages:
    2,989
    Likes Received:
    2,883
    upload_2022-9-21_12-18-2.gif

    Apply the constant multiple rule (d/dx)(c F(x)) with upload_2022-9-21_12-19-15.gif and upload_2022-9-21_12-19-46.gif

    upload_2022-9-21_12-24-29.gif = upload_2022-9-21_12-29-0.gif

    apply the chain rule (d/dx(F(G(x)))=(d/du)(F(u))(d/dx)(G(x)):

    upload_2022-9-21_12-31-38.gif = upload_2022-9-21_12-34-12.gif

    apply the power rule (d/du)(u^n)=n*u^(n-1) with n=-1

    = upload_2022-9-21_12-36-26.gif

    = upload_2022-9-21_12-38-42.gif


    Return to the old variable: upload_2022-9-21_12-39-26.gif


    = will continue ( only 10 img can be in one post)
     
    MathLover1, Sep 21, 2022
    #4
  5. akw

    MathLover1

    Joined:
    Jun 27, 2021
    Messages:
    2,989
    Likes Received:
    2,883
    = upload_2022-9-21_12-42-5.gif
    The derivative of a sum/difference is the sum/difference of derivatives:

    = upload_2022-9-21_12-43-19.gif


    The derivative of the cosine is (d/dx)(cos(x))=-sin(x):

    = upload_2022-9-21_12-44-20.gif

    Apply the constant multiple rule:


    = upload_2022-9-21_12-45-8.gif

    The derivative of the sine is (d/dx)(sin(x))=cos(x):

    = upload_2022-9-21_13-16-10.gif

    = upload_2022-9-21_13-16-50.gif


    = upload_2022-9-21_13-17-23.gif

    equal to zero

    upload_2022-9-21_13-18-8.gif

    will be zero if one factor equal to zero, so let -sin(theta)+mu[k]cos(theta)=0
     

    Attached Files:

    Last edited: Sep 21, 2022
    MathLover1, Sep 21, 2022
    #5
  6. akw

    MathLover1

    Joined:
    Jun 27, 2021
    Messages:
    2,989
    Likes Received:
    2,883
    then
    upload_2022-9-21_13-5-46.gif

    upload_2022-9-21_13-5-8.gif
     

    Attached Files:

    MathLover1, Sep 21, 2022
    #6
  7. akw

    akw

    Joined:
    Apr 9, 2022
    Messages:
    4
    Likes Received:
    0
    Thanks so much. That's great.

    What confuses me still, I suppose, is how one can "choose" to make one factor = 0, arbitrarily, if you see what I mean. It alarms my emotional brain, although I've checked, and I see that it does indeed yield a minimal T.

    So why choose that factor to be 0? (i.e. why not choose the other?)
     
    akw, Sep 21, 2022
    #7
  8. akw

    MathLover1

    Joined:
    Jun 27, 2021
    Messages:
    2,989
    Likes Received:
    2,883
    you want to find angle θ and it will be possible only if (dT)/(d(theta))=0

    since upload_2022-9-21_18-17-27.gif , you see that on right side is a product of two factors upload_2022-9-21_18-20-16.gif and upload_2022-9-21_18-20-54.gif

    then, you choose simpler factor which is upload_2022-9-21_18-21-10.gif and equal it to zero
     

    Attached Files:

    MathLover1, Sep 22, 2022
    #8
    1. Advertisements

Ask a Question

Want to reply to this thread or ask your own question?

You'll need to choose a username for the site, which only take a couple of moments (here). After that, you can post your question and our members will help you out.