## 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**

**Hint**

Velocity is a derivative of distance.

##
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__**Hint 2**

**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