Derivatives: Basic Rules

The position of a particle is given by $s(t) = \frac{(t^2+1)(2t-3)^3}{(t^4+5)^2}$ meters. Find the velocity $v(t) = s'(t)$.

The displacement of a particle is given by $s(t) = \frac{\sqrt{t} \, (t^2 + 1)}{t^{3/2} + 1}$ meters, where $t$ is in seconds. Find the velocity $v(t) = s'(t)$ at $t = 4$ seconds.

The population of a bacteria colony is modeled by $P(t) = \frac{(t^2 + 3t) \sqrt{t}}{t + 2}$ for $t \geq 0$ hours. Find the rate of change of the population at $t = 4$ hours.