Question

Find an expression for \(\frac{d}{{dt}}(u(t).(v(t) \times w(t)))\).

Short answer

\(\frac{d}{{dt}}(u(t).(v(t) \times w(t))) = u'(t).(v(t) \times w(t)) + u(t).(v'(t) \times w(t) + u(t).(v(t) \times w'(t))\)

Step-by-step solution

Step 1: Theorem 3 (differentiation formula)

Let us find the expression for

\(\frac{d}{{dt}}(u(t).(v(t) \times w(t)))\)

Using theorem 3 we get

\(\begin{array}{l}\frac{d}{{dt}}(u(t).(v(t) \times w(t)))\\ = u'(t).(v(t) \times w(t)) + u(t).(v(t) \times w(t))'\end{array}\)

Step 2: The conclusion

\(\begin{array}{l}\frac{d}{{dt}}(u(t).(v(t) \times w(t)))\\ = u'(t).(v(t) \times w(t)) + u(t).(v(t) \times w(t))'\\ = u'(t).(v(t) \times w(t)) + u(t).(v'(t) \times w(t) + v(t) \times w'(t))\\ = u'(t).(v(t) \times w(t)) + u(t).(v'(t) \times w(t) + u(t).(v(t) \times w'(t))\end{array}\)