Calculus 3 : Differentials

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #11 : Differentials

Find the differential  of the function:

Possible Answers:

Correct answer:

Explanation:

The differential of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #1431 : Partial Derivatives

Find the differential of the function:

Possible Answers:

Correct answer:

Explanation:

The differential of a function  is given by

The partial derivatives of the function are

Example Question #12 : Differentials

Find the differential of the function:

Possible Answers:

Correct answer:

Explanation:

The differential of the function  is given by

The partial derivatives are

Example Question #12 : Differentials

Find the total differential, , of the following function

Possible Answers:

Correct answer:

Explanation:

The total differential is defined as

For the function

We first find 

by taking the derivative with respect to  and treating  as a constant.

 

We then find 

by taking the derivative with respect to  and treating  as a constant.

We then substitute these partial derivatives into the first equation to get the total differential 

Example Question #13 : Differentials

Find the total differential, , of the following function

Possible Answers:

Correct answer:

Explanation:

The total differential is defined as

For the function 

We first find 

by taking the derivative with respect to  and treating  as a constant.

We then find 

by taking the derivative with respect to  and treating  as a constant.

 

We then substitute these partial derivatives into the first equation to get the total differential 

Example Question #14 : Differentials

Find the total differential, , of the following function

Possible Answers:

Correct answer:

Explanation:

The total differential is defined as

For the function 

We first find 

by taking the derivative with respect to  and treating  as a constant.

 

We then find 

by taking the derivative with respect to  and treating  as a constant.

 

We then substitute these partial derivatives into the first equation to get the total differential 

Example Question #15 : Differentials

If , calculate the differential  when moving from  to.

Possible Answers:

Correct answer:

Explanation:

The equation for  is

.

Evaluating partial derivatives and substituting, we get

Plugging in, we get

.

Example Question #16 : Differentials

If , calculate the differential  when moving from the point  to the point .

Possible Answers:

Correct answer:

Explanation:

The equation for  is

.

Evaluating partial derivatives and substituting, we get

Plugging in, we get

.

Example Question #17 : Differentials

If , calculate the differential  when moving from the point  to the point .

Possible Answers:

Correct answer:

Explanation:

The equation for  is

.

Evaluating partial derivatives and substituting, we get

Plugging in, we get

Example Question #19 : Differentials

Find the differential of the following function:

Possible Answers:

Correct answer:

Explanation:

The differential of the function is given by

The partial derivatives are

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