r(t) = cos^3 t + sin^3 t
How do you find r'(t)? Would it be -3sin^2 t + 3 cos^2 t or something else?
You need to use the chain rule.
$\displaystyle \frac{d}{dt}(\cos^3{t}) = -3\sin{t}\cos^2{t}$
$\displaystyle \frac{d}{dt}(\sin^3{t}) = 3\cos{t}\sin^2{t}$.
Therefore
$\displaystyle r'(t) = 3\cos{t}\sin^2{t} - 3\sin{t}\cos^2{t}$
$\displaystyle = 3\cos{t}\sin{t}(\sin{t} - \cos{t})$.