## A Train's Velocity

At one section of the track, a train travels in a straight line so that its distance (D) from a point on the tracks after time (t) is D = 5t^5 - t^4. Determine the train's velocity when t = 5.

Hint
Velocity is a derivative of distance.
Hint 2
The power rule for the first derivative:
$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$Velocity is a derivative of distance: $$v=\frac{dr}{dt}$$$
where $$v$$ is the instantaneous velocity, $$t$$ is time, and $$r$$ is position.
The power rule for the first derivative:
$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$Thus, $$D=5t^5-t^4$$$
$$v=(5)5t^{5-1}-(4)t^{4-1}$$$$$v=25t^4-4t^3$$$
Next, substitute $$t=5$$ :
$$v=25(5)^4-4(5)^3$$$$$v=15,625-500=15,125$$$
15,125