1. ## Cross product derivative

Find 1st and 2nd derivative.

$f(t) = (\bold a \times t \bold b) \times (\bold a + t^2 \bold b)$

$(\bold a \times t\bold b)' = (\bold a \times \bold b)$
$(\bold a + t^2 \bold b)' = 2t \bold b$

so...

$f'(t) = (\bold a \times t\bold b) \times (\bold a + t^2\bold b)' + (\bold a \times t\bold b)' \times (\bold a + t^2\bold b)$
$f'(t) = (\bold a \times t\bold b) \times (2t\bold b) + (\bold a \times \bold b) \times (\bold a + t^2\bold b)$

Here's where I'm having trouble.

$f''(t) = [(\bold a \times t\bold b) \times (2t\bold b)' + (\bold a \times t\bold b)' \times (\bold a + t^2\bold b) ] + [(\bold a \times \bold b) \times (\bold a + t^2\bold b)' + (\bold a \times \bold b)' \times (\bold a + t^2\bold b)] \;?$

Or should I use distributive property on that 2nd part?

$f''(t) = [(\bold a \times t\bold b) \times (2t\bold b)' + (\bold a \times \bold b)' \times (\bold a + t^2\bold b) ] + [(\bold a \times \bold b) \times (\bold a) + (\bold a \times \bold b) \times (t^2\bold b)]$

2. ## Re: Cross product

We may write

$f(t) = (\bold a \times t \bold b) \times (\bold a + t^2 \bold b)=$

$=t \; (\bold a \times \bold b) \times \bold a+t^3\; (\bold a \times \bold b) \times \bold b.$