Calculus 2 : First and Second Derivatives of Functions

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #221 : Derivatives

Use the chain rule to find the derivative of the function 

Possible Answers:

None of these answers

Correct answer:

Explanation:

The formula for the chain rule works as follows;

 

Setting , we have

 

 

Hence

 

.

 

(Keep in mind that the derivative of  is )

Example Question #222 : Derivatives

Find the second derivative of:  

Possible Answers:

Correct answer:

Explanation:

The derivative of secant  is:

Since the inner function of secant is not , we will need to use chain rule, which is to multiply the derivative of the inner function .

Solve for the first derivative.

To take the second derivative, we will need to use the product rule and chain rule.

The product rule is:  

To avoid confusion, let  and .  Their derivatives are then:  

The derivative of  in terms of  and   by product rule is:

Resubstitute the functions and derivative functions into the equation.

Simplify.  The answer is:

Example Question #223 : Derivatives

Find the derivative of:  

Possible Answers:

Correct answer:

Explanation:

Write the derivative of secant.

Since the inner function is , we will need to apply the chain rule to solve this derivative.  

Take the derivative of .  Do not mix this with the integration, which is .

The derivative of  is then:

The answer is:  

Example Question #224 : Derivatives

What is the second derivative of  ?

Possible Answers:

Correct answer:

Explanation:

We use the chain rule to get the first derivative:

This gives us . Now we must take the derivative again, which means we have to use the product rule *and* the chain rule. As a reminder, the product rule says that

Which means the second derivative of the original function is 

. With some simplifying we can see that this is equal to 

Example Question #225 : Derivatives

Evaluate the first and second derivative of the function

when 

Possible Answers:

 

 

 

 

Correct answer:

 

Explanation:

We must find the first and second derivatives.

To do so we use the power rule which states

As such

To find the second derivative we apply the power rule again

We find that

 and 

We then evaluate each for  and get

 

 

Example Question #1 : Applications Of Derivatives

Find the first and second derivatives of the function

Possible Answers:

Correct answer:

Explanation:

We must find the first and second derivatives.

We use the properties that

  • The derivative of    is  
  • The derivative of   is  

As such

To find the second derivative we differentiate again and use the product rule which states

Setting

  and  

we find that

As such

Example Question #226 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Example Question #227 : Derivatives

Find the derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is equal to

and was found using the following rules:

Example Question #228 : Derivatives

Find the second derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The first derivative of the function is equal to

and was found using the following rules:

The second derivative of the function is equal to

and was found using the same rules as above, along with

Example Question #5 : Derivative Rules For Sums, Products, And Quotients

Find the derivative of the function

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We find the answer using the quotient rule

and the product rule

and then simplifying.

 or . The extra brackets in the denominator are optional.

 

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