# Precalculus : Derivatives

## Example Questions

### Example Question #21 : Find The First Derivative Of A Function

Find the derivative of .

Explanation:

For any function , the first derivative  .

Therefore, taking each term of :

### Example Question #22 : Find The First Derivative Of A Function

Find the derivative of the following function:

Explanation:

First apply the power rule to the entire function, multiplying it by its exponent and then subtracting 1 from the new exponent. Then apply the chain rule, multiplying the expression obtained from the power rule by the derivative of just the term inside of the parentheses. Remember the derivative of cos(x) is -sin(x):

### Example Question #23 : Find The First Derivative Of A Function

Find the derivative of the following function

Explanation:

To find the derivative of this function, we simply need to use the Power Rule.  The Power rule states that for each term, we simply multiply the coefficient by the power to find the new coefficient.  We then decrease the power by one to obtain the degree of the new term.

For example, with our first term, , we would multiply the coefficient  by the power  to obtain the new coefficient of .  We then decrease the power by one from 4 to 3 for the new degree.  Therefore, our new term is .  We then simply repeat the process with the remaining terms.

Note that with the second to last term, our degree is 1.  Therefore, multiplying the coefficient by the power gives us the same coefficient of 8.  When the degree decreases by one, we have a degree of 0, which simply becomes 1, making the entire term simply 8.

With our final term, we technically have

Therefore, multiplying our coefficient by our power of 0 makes the whole term 0 and thus negligible.

Our final derivative then is

### Example Question #24 : Find The First Derivative Of A Function

Find the first derivative of the following function.

Explanation:

To find the first derivative, we will use the definition of the derivative which is,

.

In our case our,

and

.

Plugging these functions into our formula we get the following.

Now plugging in zero for h we are able to find the derivative of our function.

### Example Question #25 : Find The First Derivative Of A Function

Find  where .

Explanation:

In order to find the derivative we will need to use the power rule on each term. The power rule states,

.

Applying this rule we get the following.

### Example Question #26 : Find The First Derivative Of A Function

Find  if .

Explanation:

Because the original function is the quotient of 2 separate functions, we can use the Quotient Rule.

Quotient Rule:

If

,

then

.

gives us

, and .

Now

Or

.

Simplifying yields

or

.

We can factor out  to see that finally

.

Because the two terms  and  are different, no more terms can cancel.

### Example Question #27 : Find The First Derivative Of A Function

Find the first derivative of the function using the Product Rule. .

Explanation:

Using the Product Rule, which states,

we see that

.

### Example Question #28 : Find The First Derivative Of A Function

Find the derivative of the function where .

Explanation:

is a composition of 2 functions. The inner function is , and the outer function is . We can use the Chain Rule.

The Chain Rule states,

.

Therefore we get the following.

The rest is simplifying...

### Example Question #29 : Find The First Derivative Of A Function

Use any method to find the first derivative of the function where .

Explanation:

You can use the Chain Rule here,

In our case we get,

.

Remember the derivative of natural log is

.

Simplifying further we get our final answer.

### Example Question #30 : Find The First Derivative Of A Function

What is the first derivative of