Calculus 3 : Differentials

Example Questions

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Example Question #1 : Differentials

Compute the differentials for the following function.

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 #1 : Differentials

Find the total differential , , of the function

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

a

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

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 #1 : Differentials

Find the total derivative of the function:

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:

Example Question #6 : Differentials

Find the total derivative of the function:

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:

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:

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:

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