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Derivative of a Cross Product

  • Thread starter vroomba03
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  • #1
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Homework Statement


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)).


Homework Equations


d/dt(u(t) x v(t)) = (u'(t) x v(t) + u(t) x v'(t)

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.
 

Answers and Replies

  • #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.
 
  • #3
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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)) ?
 
  • #4
CAF123
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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)) ?
No, start by applying the chain rule to find $$\frac{d}{dt} u(f(t))$$
 
  • #5
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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))?
 
  • #6
CAF123
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Oh okay, so that is u'(f(t))(f'(t)) when using chain rule.
Yes.

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))?
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).
 
  • #7
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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)) ?
 
  • #8
CAF123
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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)) ?
Check the last term again. v is a function of g(t), not f(t).
 
  • #9
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Oops silly error. u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(g(t)(g'(t)) correct?
 
  • #10
CAF123
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Oops silly error. u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(g(t)(g'(t)) correct?
Correct :smile:
 
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  • #11
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Thank you a bunch!
 

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