Find an expression for the derivative (d/dt) of [r(t)·r'(t)×r''(t)] in simplified form.
I am attempting to use properties for both the dot product and cross product but I cannot seem to find the ones that will work to logically simplify the expression. Any clues?
It's a triple product:
$$r(t) \cdot (r'(t) \times r''(t))=\det(r(t),r'(t),r''(t))$$
The derivative of a determinant can be obtained by summing the determinants of the matrices obtained by differentiating the different columns, one at a time (see here), a process underlined by the use of red color for the "primes" below:
$$\underbrace{\det(r\color{red}{'}(t),r'(t),r''(t))}_{D_1}+\underbrace{\det(r(t),r'\color{red}{'}(t),r''(t))}_{D_2}+\underbrace{\det(r(t),r'(t),r''\color{red}{'}(t))}_{D_3}$$
$D_1=0$ and $D_2=0$ (because of $2$ identical columns).
As a consequence, the final result is $D_3$ that can be written under the form:
$$r(t) \cdot (r'(t) \times r'''(t))$$