# Calculus 3 : Derivatives

## Example Questions

### Example Question #41 : Derivatives

Calculate  of .

Explanation:

We begin by rewriting our function in a more convenient way,

.

This makes taking the derivative a little easier.  We can therefore write

, or .

### Example Question #231 : Calculus 3

Calculate  of .

Explanation:

By the chain rule, we can show

, or in this case,

.

### Example Question #232 : Calculus 3

Find  given

Explanation:

This derivative must be found using logarithmic differentiation.  Consider the following

### Example Question #233 : Calculus 3

Find  given

Explanation:

This derivative is most easily done by logarithmic differentiation.  Consider the following

### Example Question #234 : Calculus 3

Find the derivative of:

Explanation:

Take the derivative of the cosine function.  This will also require chain rule, which is the derivative of the inner function.

### Example Question #235 : Calculus 3

Find the derivative:

Explanation:

This problem will require multiple chain rule.  Take the derivative of natural log, and then apply the inner function of the natural log, cosine, and then also apply the chain rule for .

Combine this into one term and simplify.

### Example Question #236 : Calculus 3

Evaluate

Explanation:

Since we are taking the (full) derivative of an expression involving  and , we must use implicit differentiation. We proceed as follows

start.

. Using , and the Product Rule.

. Factor

### Example Question #237 : Calculus 3

Evaluate the limit

by interpreting the limit in terms of the definition of the derivative.

The limit does not exist.

Explanation:

The definition of the derivative is

.

Equating

, to the definition of the derivative, we have

.

### Example Question #238 : Calculus 3

What is the derivative of ?

Explanation:

Step 1: Take the derivative of the first term:

via the power rule.
Step 2: Take the derivative of the next term:
, again using the product rule.
Step 3: Take the derivative of the last term .

We will have to use Quotient Rule here.

Find the derivative of f(x) and g(x):

.

Use the formula:

.

Step 4: Put all the final terms from highest to lowest degree...

We get:.

### Example Question #239 : Calculus 3

Find the derivative of