# Calculus 3 : Applications of Partial Derivatives

## Example Questions

### Example Question #51 : Applications Of Partial Derivatives

Find  of the function:

Explanation:

The gradient 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 #52 : Applications Of Partial Derivatives

Find  of the function

Explanation:

The formula for the gradient of F is

.

Using the rules for partial differentiation, we get

.

Putting into vector notation, we get

### Example Question #53 : Applications Of Partial Derivatives

Find the equation of the plane tangent to the point  if the gradient vector .

Explanation:

By definition,  is the vector that orthogonal to the plane at the point we were given.

We then use the formula for a plane given a point  and normal vector .

We get

.

Through algebraic manipulation, we get

.

### Example Question #54 : Applications Of Partial Derivatives

Find the gradient, , of the function .

Explanation:

The gradient of a function  is as follows:

.

We compute the derivative of the function with respect to each of the variables and treat the others like constants.

Using the rule

,

we obtain

,

and

.

Putting these expressions into the vector completes the problem, and you obtain

.

### Example Question #55 : Applications Of Partial Derivatives

Find the gradient, , of the function .

Explanation:

The gradient of a function  is as follows:

.

We compute the derivative of the function with respect to each of the variables and treat the others like constants.

Using the rule

,

we obtain

,

and .

Putting these expressions into the vector completes the problem, and you obtain

.

### Example Question #56 : Applications Of Partial Derivatives

Find the gradient, , of the function .

Explanation:

The gradient of a function  is as follows:

.

We compute the derivative of the function with respect to each of the variables and treat the others like constants.

Using the rule

,

we obtain

,

and

.

Putting these expressions into the vector completes the problem, and you obtain

### Example Question #57 : Applications Of Partial Derivatives

Find  of the function .

Explanation:

The gradient of a function  is as follows:

.

We compute the derivative of the function with respect to each of the variables and treat the others like constants.

Using the rule , we obtain

,

and

.

Putting these expressions into the vector completes the problem, and you obtain

.

### Example Question #58 : Applications Of Partial Derivatives

Find the tangent plane to the surface given by

at the point

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. 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:

Now, we evaluate them at the given point:

Finally, plug in all of our information into the formula and simplify:

### Example Question #59 : Applications Of Partial Derivatives

Find the tangent plane to the surface given by

at the point

Explanation:

The equation of the tangent plane is given by

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

The partial derivatives are

The rules used to find the derivatives are

Evaluated at the given point, the partial derivatives are

Plugging all of this into the above formula, and simplifying, we get

### Example Question #60 : Applications Of Partial Derivatives

Find the tangent plane to the surface given by

at the point

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. 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:

Next, we evaluate the partial derivatives at the given point:

Plugging in all our information into the formula above, we get