Calculus 3 : Differentials

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Differentials

Compute the differentials for the following function.

Possible Answers:

Correct answer:

Explanation:

What we need to do is take derivatives, and remember the general equation.

When taking the derivative with respect to y recall that the product rule needs to be used.

Example Question #2 : Differentials

Find the total differential , , of the function

Possible Answers:

Correct answer:

Explanation:

The total differential is defined as

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 #3 : Differentials

Find the total differential, , of the function

Possible Answers:

a

Correct answer:

Explanation:

The total differential is defined as

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 #4 : Differentials

Find the total differential, , of the function

Possible Answers:

Correct answer:

Explanation:

The total differential is defined as

We first find  by taking the derivative with respect to  and treating the other variables as constants.

We then find  by taking the derivative with respect to  and treating the other variables as constants.

We then find  by taking the derivative with respect to  and treating the other variables as constants.

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

Example Question #5 : Differentials

Find the total derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The total derivative of a function of two variables is given by the following:

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

The partial derivatives of the function are

The derivatives were found using the following rules:

So, our final answer is

 

Example Question #6 : Differentials

Find the total derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The total derivative of a function of two variables 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

The derivatives were found using the following rules:

Example Question #7 : Differentials

Find the total derivative of the function:

Possible Answers:

Correct answer:

Explanation:

The total derivative of a function  is given by

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

The partials are

The derivatives were found using the following rules:

 

 

Example Question #8 : Differentials

Find the total derivative of the following function:

Possible Answers:

Correct answer:

Explanation:

The total derivative of a 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

The derivatives were found using the following rules:

Example Question #9 : Differentials

Find the differential of the following function:

Possible Answers:

Correct answer:

Explanation:

The differential of a function is given by

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

The partial derivatives are

The derivatives were found using the following rules:

Example Question #10 : Differentials

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

The derivatives were found using the following rules:

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