Given the position function s(t) = t cos t, find the velocity function

v(t) = ds/dt = 1*cost - t*sint

using the product rule

To find the velocity function, we need to take the derivative of the position function with respect to time.

Given the position function s(t) = t cos t, we can differentiate it with respect to time (t) to find the velocity function.

To differentiate the function, we will employ the product rule and chain rule.

Product Rule:
(d/dt) [f(t) * g(t)] = f'(t) * g(t) + f(t) * g'(t)

Chain Rule:
(d/dt) [f(g(t))] = f'(g(t)) * g'(t)

Let's differentiate each term in the position function:

d/dt [t] = 1 (derivative of t with respect to t is 1)

For the second term, d/dt [cos t]:

The derivative of cos t with respect to t is -sin t

Now, let's apply the product rule to differentiate s(t) = t cos t:

s'(t) = (d/dt) [t cos t]
= (d/dt) [t] * cos t + t * (d/dt) [cos t]
= 1 * cos t + t * (-sin t)
= cos t - t sin t

Therefore, the velocity function v(t), which represents the derivative of the position function, is:

v(t) = cos t - t sin t