# Calculus 3 : Multi-Variable Chain Rule

## Example Questions

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### Example Question #1 : Multi Variable Chain Rule

Compute  for .

Explanation:

All we need to do is use the formula for multivariable chain rule.

When we put this all together, we get.

### Example Question #2 : Multi Variable Chain Rule

Evaluate  in terms of  and/or  if ,  , and .

Explanation:

Expand the equation for chain rule.

Evaluate each partial derivative.

,

Substitute the terms in the equation.

Substitute .

### Example Question #3 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem

### Example Question #4 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem

### Example Question #5 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem

### Example Question #6 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem

### Example Question #7 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem

### Example Question #1 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem

### Example Question #1 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem,

### Example Question #10 : Multi Variable Chain Rule

Use the chain rule to find  when  .

Explanation:

The chain rule states .

Since  and  are both functions of  must be found using the chain rule.

In this problem,

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