vector function derivative help

Question

I need help with finding the derivative of a vector function, we haven't done any examples in class hence I have no idea how to proceed. So we have

$\alpha:[a, b] \to R^2, \alpha'(t) \neq (0,0) $

Find the derivative of

$\phi(t) = ln (||\alpha(t)||) + <b, \alpha(t)> $

where $b := (2,3)$

Thanks a lot in advance.

Answer 1

We have $\alpha(t)=(\alpha_1(t),\alpha_2(t))$, where $\alpha_1(t)$ maps $[a,b]$ on the $x$-axis and $\alpha_2(t)$ maps $[a,b]$ on the $y$-axis. Write

$$ \phi(t)=\ln\left(\sqrt{[\alpha_1(t)]^2+[\alpha_2(t)]^2}\right)+2\alpha_1(t)+3\alpha_2(t) $$

and use the chain rule to take $\frac{d}{dt}$ in the usual manner (since $\phi$ is not a vector valued function).