Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #240 : Calculus 3

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative can be taken by taking the individual derivatives of each component of the function. The first component requires the use of the product rule, which states that:

Following this rule, the derivative of the first component is:

The derivatives were found by using the following rules:

Then, the derivative of the last component is found by using the power rule which states that:

The derivative of the final component is 

With each individual component found, they can be added together to get the final answer:

Example Question #51 : Derivatives

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 product rule which states that:

Using this rule, the derivative of the function is:

The derivatives were found by using the following rules:

After some simplification, the final answer is:

Example Question #52 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The derivative can be taken by taking the individual derivatives of each component of the function. The first component's derivative can be found by using the chain rule, which states that:

In this case, the derivative of the first component is:

This derivative was found by also using the fact that:

The derivative of the second component of the function can be found by using the quotient rule, which states that:

Following this rule, the derivative of the second component of the function is 

Once all of the derivatives have been found , they can be added together to get the final answer which is:

 

 

Example Question #53 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The only way to take the derivative of the function is by using the product rule which states that:

Using this rule, the derivative of the function is:

The derivatives were found by also using the following rules:

Example Question #54 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

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

Therefore, the derivative of the function is:

The derivative was found by also using the following rules:

Example Question #55 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The only way to find the derivative of the function is by using the rule for differentiating a natural log function which states that:

In order to solve for the u' portion, you must use the quotient rule which states that:

You must also use the following rules when expressing u':

Therefore:

When combining this with the rest of the formula for differentiating natural log functions, you get:

After some rewriting and cancellation, the final answer is:

 

 

Example Question #56 : 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:

You will also need the following rules to evaluate the exponential and quadratic components of the function:

Therefore, the final answer is:

Additionally, the denominator of the derivative was found by using the following rule:

Example Question #57 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The only way to find the derivative of this function is by knowing how to differentiate exponential functions. You must know that:

Therefore, the first part of the derivative of this function is the original function: 

Next, you must take the derivative of the exponent. The only way to do this is by using the chain rule which states that:

Therefore, the derivative of the exponent is:

This derivative was also found by using the following rule:

With each component found, the final answer is:

Example Question #58 : Derivatives

Find the derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

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

Therefore, part of the derivative contains the original function. However, solving for u' requires the following formula:

Combining all of these rules, the final answer is:

Example Question #59 : Derivatives

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 product rule, which states that:

Following this rule, you encounter the fact that you must differentiate a trig function with an exponent greater than one. To do this, you must use the chain rule which states that:

Therefore, the derivative of the aforementioned trig function is:

The sine function came from the fact that 

To find the derivative of the remainder of the function, you must also use the following rule:

Combining all of these elements, the final answer is:

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