# Calculus 3 : Derivatives

## Example Questions

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### Example Question #1 : How To Find Velocity

The position of a particle is given by .  Find the velocity at .

Explanation:

The velocity is given as the derivative of the position function, or

.

We can use the quotient rule to find the derivative of the position function and then evaluate that at .  The quotient rule states that

.

In this case,  and .

We can now substitute these values in to get

.

Evalusting this at  gives us .

### Example Question #1 : Derivatives

The position of an object is given by the following equation:

Determine the equation for the velocity of the object.

Explanation:

Velocity is the derivative of position, so in order to find the equation for the velocity of an object, all we must do is take the derivative of the equation for its position:

We will use the power rule to get the derivative.

Therefore we get,

### Example Question #11 : Spatial Calculus

The position of a particle is represented by . What is the velocity at ?

Explanation:

Differentiate the position equation, to get the velocity equation

Now we plug 4 into the equation to find the velocity

is approximately equal to 2.72. Therefore

### Example Question #3 : Derivatives

What is the velocity function when the position function is given by

.

Explanation:

To find the velocity function, we need to find the derivative of the position function.

So lets take the derivative of  with respect to .

The derivative of  is  because of Power Rule:

The derivative of  is  due to Power Rule

So...

### Example Question #41 : How To Find Velocity

Consider the position function , which describes the positon of an oxygen molecule.

Find the function which models the velocity of the oxygen molecule.

Explanation:

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

So given:

Apply the power rule to each term to find the velocity.

### Example Question #5 : Derivatives

The position of an object is given by the equation . What is the velocity of the object at ?

Explanation:

The velocity of the object can be found by differentiating the position equation of the object. To differentiate the position equation of the object, we can use the power rule for the second term where if

Using this rule we find that

We can now use the value of  to solve for the velocity at

### Example Question #1 : Derivatives

The position of an object is given by the equation . What is the equation for the velocity of the object?

Explanation:

The velocity of the object can be found by differentiating the position equation. The position equation can be accurately differentiated using the power rule and the product rule where if

and where if

Using these two rules we find the velocity equation to be

### Example Question #8 : How To Find Acceleration

The position of a particle is given by .  Find the acceleration of the particle when .

Explanation:

The acceleration of a particle is given by the second derivative of the position function.  We are given the position function as

.

The first derivative (the velocity) is given as

.

The second derivative (the acceleration) is the derivative of the velocity function.  This is given as

.

Evaluating this at  gives us the answer.  Doing this we get

.

### Example Question #2 : Derivatives

The position of an object is described by the following equation:

Find the acceleration of the object at  second.

Explanation:

Acceleration is the second derivative of position, so we must first find the second derivative of the equation for position:

Now we can plug in t=1 to find the acceleration of the object after 1 second:

### Example Question #2 : Derivatives

If  models the distance of a projectile as a function of time, find the acceleration of the projectile at .