Multivariable Calculus : Partial Differentiation

Study concepts, example questions & explanations for Multivariable Calculus

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Example Questions

Example Question #1 : Differentiation Rules

Find .

Possible Answers:

Correct answer:

Explanation:

In order to find , we need to take the derivative of  in respect to , and treat , and  as constants. We also need to remember what the derivatives of natural log, exponential functions and power functions are for single variables.

Natural Log:

Exponential Functions:

Power Functions:

 

Example Question #2 : Differentiation Rules

Find .

Possible Answers:


Correct answer:

Explanation:

In order to find , we need to take the derivative of  in respect to  , and treat , and  as constants. We also need to remember what the derivatives of natural log, exponential functions and power functions are for single variables.

Natural Log:

Exponential Functions:

Power Functions:

Example Question #3 : Differentiation Rules

Find .

Possible Answers:

Correct answer:

Explanation:

In order to find , we need to take the derivative of  in respect to , and treat , and  as constants. We also need to remember what the derivatives of natural log, exponential functions and power functions are for single variables.

Natural Log:

Exponential Functions:

Power Functions:

 

Example Question #1 : Arc Length

Determine the length of the curve , on the interval .

Possible Answers:

Correct answer:

Explanation:

First we need to find the tangent vector, and find its magnitude.

 

Now we can set up our arc length integral

 

 

Example Question #2 : Partial Differentiation

Determine the length of the curve , on the interval .

Possible Answers:

Correct answer:

Explanation:

First we need to find the tangent vector, and find its magnitude.

 

Now we can set up our arc length integral

 

 

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