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Derivative of a Cross Product

  1. Sep 9, 2013 #1
    1. The problem statement, all variables and given/known data
    Assume that you are given differentiable function f(t) and g(t). Find a formula for the
    derivative of the cross product u(f(t)) x v(g(t)).


    2. Relevant equations
    d/dt(u(t) x v(t)) = (u'(t) x v(t) + u(t) x v'(t)

    3. The attempt at a solution
    So in this case I was thinking that you would just substitute f(t) and g(t) where t would be in the regular equation, so it would be U'(f(t)) x v(g(t)) + u(f(t)) x v'(g(t)), for the equation, but I have a feeling that thats not right just because it seems too simple.
     
  2. jcsd
  3. Sep 9, 2013 #2

    CAF123

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    u'(f(t)) = d u(f(t))/f(t) and similar result for v'(g(t)).
    What you want is d/dt ( u(f(t)) × v(g(t)) ), so you will have to use chain rule.
     
  4. Sep 9, 2013 #3
    So then the equation would be U'(du(f(t))/f(t)) x V(g(t)) + U(g(t)) x V'(du(g(t))/g(t)) x U(f(t)) ?
     
  5. Sep 9, 2013 #4

    CAF123

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    No, start by applying the chain rule to find $$\frac{d}{dt} u(f(t))$$
     
  6. Sep 9, 2013 #5
    Oh okay, so that is u'(f(t))(f'(t)) when using chain rule. So you said u'(f(t)) = d u(f(t))/f(t), so then would I divide what i got by the chain rule and divide it by f(t) and that would be my u'(f(t))?
     
  7. Sep 9, 2013 #6

    CAF123

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    Yes.

    You need to find $$\frac{d}{dt} \left(u(f(t)) \times v(g(t))\right) = \frac{d}{dt} u(f(t)) \times v(g(t)) + u(f(t)) \times \frac{d}{dt} v(g(t))$$

    You correctly found ##\frac{d}{dt} u(f(t))##. Now find ##\frac{d}{dt} v(g(t))## and substitute in.

    What I should have wrote is u'(f(t)) ##\equiv## d u(f(t))/f(t), these two expressions denote the derivative of u with respect to f(t).
     
  8. Sep 9, 2013 #7
    So the final equation for the question would be u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(f(t)(f'(t)) ?
     
  9. Sep 9, 2013 #8

    CAF123

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    Check the last term again. v is a function of g(t), not f(t).
     
  10. Sep 9, 2013 #9
    Oops silly error. u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(g(t)(g'(t)) correct?
     
  11. Sep 9, 2013 #10

    CAF123

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    Correct :smile:
     
  12. Sep 9, 2013 #11
    Thank you a bunch!
     
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