Derivative of Cross Product with Differentiable Functions

[vroomba03]

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 that's not right just because it seems too simple.

 

[CAF123]

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.

 

[vroomba03]

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)) ?

 

[CAF123]

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 $$\frac{d}{dt} u(f(t))$$

 

[vroomba03]

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))?

 

[CAF123]

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).

 

[vroomba03]

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)) ?

 

[CAF123]

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).

 

[vroomba03]

Oops silly error. u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(g(t)(g'(t)) correct?

 

[CAF123]

Oops silly error. u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(g(t)(g'(t)) correct?

Correct

 

[vroomba03]

Thank you a bunch!