# Derivative of a Cross Product

## 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.

CAF123
Gold Member
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.

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

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
Gold Member
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).

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
Gold Member
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).

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

CAF123
Gold Member
Oops silly error. u'(f(t))(f'(t)) x v(g(t)) + u(f(t)) x v'(g(t)(g'(t)) correct?
Correct

1 person
Thank you a bunch!