# Calculus 3 : Calculus Review

## Example Questions

### Example Question #244 : How To Find Acceleration

The position of a particle is given by the function . What is the acceleration of the particle at time ?

Explanation:

Acceleration of a particle can be found by taking the derivative of the velocity 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 velocity with respect to time, we are evaluating how velocity changes over time; i.e acceleration! This is just like finding velocity by taking the derivative of the position function.

To take the derivative of the 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

And the acceleration function is

At time

### Example Question #258 : How To Find Acceleration

The position at a certain point is given by:

What is the acceleration at ?

Explanation:

In order to find the acceleration of a given point, you must first find the derivative of the position function which gives you the velocity function:

Then, you differentiate the velocity function to get the acceleration function:

Then, find  when .

### Example Question #21 : Calculus Review

The position of a certain point is given by the following function:

What is the acceleration at ?

Explanation:

In order to find the acceleration of a certain point, you must first find the derivative of the position function which gives us the velocity function and then the derivative of the velocity function which gives us the acceleration function:

In this case, the position function is:

The velocity function is found by taking the derivative of the position function:

The acceleration function is found by taking the derivative of the velocity function:

Finally, to find the accelaration, substitute  into the acceleration function:

### Example Question #1 : Computation Of Derivatives

Give .

Explanation:

, and the derivative of a constant is 0, so

### Example Question #1 : Computation Of Derivatives

Give .

Explanation:

First, find the derivative  of .

, and the derivative of a constant is 0, so

Now, differentiate  to get .

Differentiate .

Explanation:

, so

### Example Question #1 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Give .

Explanation:

First, find the derivative  of .

Recall that , and the derivative of a constant is 0.

Now, differentiate  to get .

### Example Question #2 : First And Second Derivatives Of Functions

Find the second derivative of the following function.

Explanation:

To find the second derivative, first we need to find the first derivative. So for the given function, we get the first derivative to be

Now we have to take the derivative of the derivative. To do this we need to use the product rule as shown below

Thus, we get

.

### Example Question #6 : First And Second Derivatives Of Functions

Find the derivative of .

Explanation:

To solve this derivative, we need to use logarithmic differentiation. This allows us to use the logarithm rule  to solve an easier derivative.

Let .

Now we'll take the natural log of both sides to get

.

Now we can use implicit differentiation to solve for .

The derivative of  is , and the derivative of  can be found using the product rule, which states

where  and  are functions of .

Letting  and

(which means  and ) we get our derivative to be .

Now we have , but , so subbing that in we get

.

Multiplying both sides by , we get

.

That is our derivative.

### Example Question #1 : First And Second Derivatives Of Functions

The position of a car is given by the following function:

What is the velocity function of the car?