# The position of an object moving along a line is given by p(t) = 2t - sin(( pi )/6t) . What is the speed of the object at t = 3 ?

Speed $p ' \left(3\right) = 2$

#### Explanation:

Given the position equation $p \left(t\right) = 2 t - \sin \left(\frac{\pi t}{6}\right)$
The speed is the rate of change of the position p(t) with respect to t.

We calculate the first derivative at t=3

$p ' \left(t\right) = \frac{d}{\mathrm{dt}} \left(2 t - \sin \left(\frac{\pi t}{6}\right)\right)$

$p ' \left(t\right) = \frac{d}{\mathrm{dt}} \left(2 t\right) - \frac{d}{\mathrm{dt}} \sin \left(\frac{\pi t}{6}\right)$

$p ' \left(t\right) = 2 - \left(\frac{\pi}{6}\right) \cdot \cos \left(\frac{\pi t}{6}\right)$

at $t = 3$
$p ' \left(3\right) = 2 - \left(\frac{\pi}{6}\right) \cdot \cos \left(\frac{\pi \cdot 3}{6}\right)$

$p ' \left(3\right) = 2 - 0$

$p ' \left(3\right) = 2$

God bless....I hope the explanation is useful.