Calculus 2 : Derivative Review

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #371 : Derivatives

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The first term of the function's derivative is found simply with the power rule:

The second term is a trig derivative with the chain rule:

The third term requires the rule for derivatives of exponential and the chain rule:

Combining these three terms, we get the final answer:

Example Question #171 : First And Second Derivatives Of Functions

Compute the first derivative of the following function using the product rule:

Possible Answers:

Correct answer:

Explanation:

The rule for the derivative of the product of two functions is as follows:

For the expression given, the two functions are:

Whose derivatives are:

The derivative of the product is then:

Simplified, this reduces to:

Example Question #173 : First And Second Derivatives Of Functions

Compute the first derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The rule for the derivative of the product of two functions is as follows:

For the expression given, the two functions are:

Whose derivatives are:

The derivative of the product is then:

Simplifying, we get:

Example Question #1494 : Calculus Ii

Find the first derivative of ?

Possible Answers:

Correct answer:

Explanation:

Step 1: Use the power rule on the first term...

 becomes 

Step 2: Take derivative of second term...

 becomes 

Note: The derivative of a(n) constant term is always zero.

The first derivative is 

Example Question #175 : First And Second Derivatives Of Functions

What is the first derivative of ?

Possible Answers:

Correct answer:

Explanation:

Step 1: Find the derivative of  and , which we denote as  and .



Step 2: Plug in the functions into the quotient rule formula:



Step 3: Simplify...



Step 4: Reduce:



The first derivative is 

Example Question #171 : First And Second Derivatives Of Functions

Find  if ?

Possible Answers:

Correct answer:

Explanation:

Step 1: We just take the derivative of , which is given to us in the question.



Step 2: Use the power rule to take the derivative..

First Term: 
Second Term: 

Step 3: Take the final answers in Step 2 and add them together...

The second derivative, , is 

Example Question #371 : Derivative Review

Find the first derivative of: 

Possible Answers:

Correct answer:

Explanation:

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Example Question #372 : Derivative Review

Find the first derivative of: .

 

Possible Answers:

Correct answer:

Explanation:

Recall that when taking the derivative, multiply the exponent by the coefficient in front of the x term and then subtract one from the exponent:

Example Question #174 : First And Second Derivatives Of Functions

What is the first derivative of 

Possible Answers:

Correct answer:

Explanation:

Step 1: Re-write  as  to a power:


Step 2: Use the power rule:



Step 3: Simplify the exponent:



Step 4: Re-write  as a fraction with a positive exponent:


Step 5: Multiply the result in Step 4 with the coefficient  in step 3:

 

Example Question #373 : Derivative Review

Possible Answers:

Correct answer:

Explanation:

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