Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Product rule: 

Note that u and v may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Quotient rule: 

Note that u and v may represent large functions, and not just individual variables!

Using the above properties, the velocity function is

At time 

The velocity function is given by the derivative of the position function. Use the power rule to find the derivative.

The power rule is given by:

Applying the power rule to this equation, remembering to multiply by the constants in each term:

Solve for the velocity at time  by plugging  into the velocity equation to recieve the correct answer, .

To find the derivatives in this particular problem use the power rule which states,

The velocity is simply the first derivative of the position function.

Apply the power rule to find the following derivative.

At time (t=3 seconds):

This is one of the answer choices.

To find the derivative in this particular problem use the power rule which states,

Given the position function, the velocity is simply the first derivative:

At time (t=2 seconds):

This is one of the answer choices.

In order to find the velocity function from the position function, we must find the derivative of the position function since

.

Because the derivative of the exponential function is the exponential function itself, we find that

.

Thus, we find that

.

If p(t) models the position of a particle as a function of time, find the function which models the particle's velocity.

Recall that velocity is the first derivative of position (acceleration is the second derivative of position).

So, we must find the first derivative of p(t). Remember, finding derivatives of polynomials is as easy as multiplying by the exponent and decreasing the current exponent by 1.

In other words...

Note how each term "lost" a t, and how the old exponent is now out in front. Furthermore, don't forget to lose the constant term (-18) 

So our answer is

If the function  models the acceleration of a ball falling down a flight of stairs, find the velocity of the ball when .

We can begin this problem by recalling that acceleration is the derivative of velocity. Therefore, we need to integrate our function to find our velocity function.

Making our velocity function:

But, we still need v(5)...

Quite the velocity indeed. However, since no units are provided, we can assume we are dealing with very small units...

If the function b(s) models the acceleration of a ping pong ball being thrown across a table, find the function which models the ball's velocity if the velocity when  is 108.

We are asked to relate acceleration and position. This may seem confusing, but if we remember that acceleration is the first derivative of velocity, we know that we should start by integrating b(s).

Now we have the following:

Which is almost our velocity function, but we need to use our given conditions to find c.

We are told that when s=3, the velocity is equal to 108. Let's plug those numbers in and solve for c!

Which makes our final answer:

To find the velocity function of the motorbike, take the derivative of the position curve.

Substitute the time after three seconds: