The velocity function for the spaceship is the first derivative of the position function of the spaceship:

and was found using the following rules:

, 

, 

.

The velocity function is the integral of the accleration function. So, we must integrate the given function:

The integration was performed using the following rule:

Now, solve for C by plugging in the initial condition:

Our final answer is

The velocity function is given by the first derivative of the position function:

and was found using the following rules:

, 

, 

, 

To find the answer to this problem it's essential to realize that velocity is just the anti-derivative of acceleration. So if we take the indefinite integral of our acceleration function, we should get the function for our velocity.

We know that 

,

and in specific, the anti-derivative of a constant  is , where  is just an arbitrary constant.

So, 

We have a new problem. We need to figure out what this C is. It must be our initial velocity because . However, the brick was merely dropped from the roof of the brick factory and not thrown so it didn't start out with any velocity at all. So we can say our C is 0.

Now to figure out the answer to our question, we just have to put  into the velocity function.

This is a simple calculation:

.

Remember that velocity  is the derivative of position  with respect to time 

.

Since our position function has a function within a function, we must utilize the chain rule to differentiate with respect to .

By the chain rule, we know that

In our case, , ,  by the trigonometric identity, and  by the power rule. 

Since the question wanted the velocity at ,

Since cosine is an even function,

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

 .

When taking the derivative, we will use the power rule which states

and by applying this rule to each term we get

.

Hence,

.

If you're give the position function, you take the derivative of that to find the velocity function.

To take the derivative, multiply the exponent by the coefficient to get your new coefficient and then decrease the exponent by 1. (Remember, constants have a derivative of 0!).

In this case, the velocity function is

.

Once you find that function, you plug in 1 for x.

That becomes .

Therefore, the answer is 7.

The velocity function is the derivative of the position function.

Therefore, take the derivative of the position function given. (Multiply the exponent of each term by the coefficient and also decrease the exponent by 1 to get the derivative. Constants have a derivative of 0!)

The velocity function is:

.

To find the velocity of a position function, just take the derivative. Remember, to take the derivative, multiply the exponent by the coefficient of a term and then also take one away from the exponent, this is known as the power rule.

(Constants have a derivative of 0).

Therefore, the derivative function is:

.

Then, plug in 1 for t.

Therefore,

.

Velocity is the time derivative of position:

In the case of this problem, there are two position functions describing two-dimensional position:

This means there are two velocity functions to be found describing two-dimensional motion:

At time 

The magnitude of the velocity is then found by taking the root of the sum of squares of the velocities: