Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #231 : Derivative Review

Find the derivative of the following function at :

Possible Answers:

Correct answer:

Explanation:

To find the derivative, we must use the following rule:

Now, using the above rule, write out the derivative:

The internal derivatives were found using the following rules:

Evaluated at , we get

 

Example Question #232 : Derivative Review

Find the derivative of the function .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

We will need to use the product rule and the chain rule to find the derivative.

. Start

. Product Rule

. Use the Chain Rule for .

. Multiply

. Factor out a , and an .

Example Question #233 : Derivative Review

Find the derivative of the function 

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Although we could use the Product Rule to compute the derivative, it becomes much easier to find if we rewrite .

 

. Start

.

Example Question #11 : Derivative Rules For Sums, Products, And Quotients

If , find  in terms of  and .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

Using a combination of logarithms, implicit differentiation, and a bit of algebra, we have

. Quotient Rule + implicit differentiation.

Example Question #12 : Derivative Rules For Sums, Products, And Quotients

Find the derivative of the function 

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is .

 

Using the Quotient Rule and the fact , we have

 

.

Example Question #234 : Derivative Review

Find the derivative of the function 

Possible Answers:

None of the other answers

Correct answer:

Explanation:

The Chain Rule is required here.

. Start

 

The Chain Rule Proceeds as follows: . In this case

 

and

 .

Putting these into the Chain Rule, we get

Or the same to say

.

Example Question #235 : Derivative Review

Evaluate the derivative of , where is any constant.

Possible Answers:

None of the other answers

Correct answer:

Explanation:

For the term, we simply use the power rule to abtain . Since is a constant (not a variable), we treat it as such. The derivative of any constant (or "stand-alone number") is .

Example Question #236 : Derivative Review

Find the derivative of the function .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

We use the Product Rule to find our answer here. The Product Rule formula is .

Let , , then we have , .

Putting these into our formula, we have

.

Example Question #237 : Derivative Review

What is the derivative of 

?

Possible Answers:

Correct answer:

Explanation:

We can find the derivative of 

using the power rule

with 

so we have

Example Question #238 : Derivative Review

Find the velocity function given the displacement function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the displacement function is the velocity, so we need to find . We can use the power rule 



with 

to get 

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