Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #41 : Derivatives

Calculate  of .

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

Explanation:

By the chain rule, we can show

, or in this case,

.

Example Question #232 : Calculus 3

Find  given 

Possible Answers:

Correct answer:

Explanation:

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

Example Question #233 : Calculus 3

Find  given 

Possible Answers:

Correct answer:

Explanation:

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

Example Question #234 : Calculus 3

Find the derivative of:  

Possible Answers:

Correct answer:

Explanation:

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

The answer is:  

Example Question #235 : Calculus 3

Find the derivative:  

Possible Answers:

Correct answer:

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.

The answer is:  

Example Question #236 : Calculus 3

Evaluate 

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is .

 

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.

Possible Answers:

The limit does not exist.

Correct answer:

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 ?

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

Explanation:

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