Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #61 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The easiest way to find the derivative of this function is by using the quotient rule which states that:

To find the derivative of the numerator, you must use the power rule which states that:

To find the derivative of the denominator, you must use the chain rule which states that:

After applying all of the stated rules, you get that the derivative of the function is:

After further simplifications, the final answer is:

 

Example Question #251 : Calculus 3

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The only way to find the derivative of this function is by using the following rule:

Therefore, part of the derivative will contain the original function, . To solve for the u' portion of the equation, you must take the derivative of the exponent which requires the use of the rule again. The derivative of the exponent is:

The cosecant comes from the following rule:

After combining all of the components in accordance with the rule for differentiating exponential functions, the final answer is:

The simplification was made using the following rule:

Example Question #252 : Calculus 3

Find  if 

Possible Answers:

Correct answer:

Explanation:

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Example Question #253 : Calculus 3

What is the derivative of ?

Possible Answers:

Correct answer:

Explanation:

Use Chain Rule. Identify  and 



Find the derivatives:



Use the formula: 

 

Example Question #254 : Calculus 3

What is the derivative of 

Possible Answers:

None of the Above

Correct answer:

Explanation:

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Example Question #255 : Calculus 3

Find the first derivative of .

Possible Answers:

None of the Above

Correct answer:

Explanation:

Take the derivative of the function...

The derivative of  is , or any power .

We will take the derivative of the exponent and drag it down to the coefficient:

Example Question #256 : Calculus 3

Find the derivative of .

Possible Answers:

None of the Above

Correct answer:

Explanation:

Find the derivative of each term:

The derivative of  is .
The derivative of  is .
The derivative of  is .

The derivative of  is .

The derivative of 
Put all of the derivatives together...

Example Question #257 : Calculus 3

Find the derivative of 

Possible Answers:

None of the Above

Correct answer:

Explanation:

Separate  and :


Take the derivative of each function:



Use the Product Rule Formula:

Expand and simplify:



The derivative of the function is 

Example Question #258 : Calculus 3

What is the derivative of ?

Possible Answers:

None of the Above

Correct answer:

Explanation:

Take the derivative of each term. Use the power rule:



Add up all the derivatives...

Final Answer is 

Example Question #259 : Calculus 3

Find the derivative of .

Possible Answers:

None of the Above

Correct answer:

Explanation:

Use the power rule to find the derivatives:


Add both derivatives together:

 

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