# Calculus Examples

Step 1

Step 1.1

Differentiate.

Step 1.1.1

By the Sum Rule, the derivative of with respect to is .

Step 1.1.2

Differentiate using the Power Rule which states that is where .

Step 1.2

Evaluate .

Step 1.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.2.3

Multiply by .

Step 2

Step 2.1

By the Sum Rule, the derivative of with respect to is .

Step 2.2

Evaluate .

Step 2.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.2.2

Differentiate using the Power Rule which states that is where .

Step 2.2.3

Multiply by .

Step 2.3

Evaluate .

Step 2.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.3.2

Differentiate using the Power Rule which states that is where .

Step 2.3.3

Multiply by .

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to and solve.

Step 4

Step 4.1

Find the first derivative.

Step 4.1.1

Differentiate.

Step 4.1.1.1

By the Sum Rule, the derivative of with respect to is .

Step 4.1.1.2

Differentiate using the Power Rule which states that is where .

Step 4.1.2

Evaluate .

Step 4.1.2.1

Since is constant with respect to , the derivative of with respect to is .

Step 4.1.2.2

Differentiate using the Power Rule which states that is where .

Step 4.1.2.3

Multiply by .

Step 4.2

The first derivative of with respect to is .

Step 5

Step 5.1

Set the first derivative equal to .

Step 5.2

Factor out of .

Step 5.2.1

Factor out of .

Step 5.2.2

Factor out of .

Step 5.2.3

Factor out of .

Step 5.3

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Step 5.4

Set equal to .

Step 5.5

Set equal to and solve for .

Step 5.5.1

Set equal to .

Step 5.5.2

Solve for .

Step 5.5.2.1

Add to both sides of the equation.

Step 5.5.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

Step 5.5.2.3

The complete solution is the result of both the positive and negative portions of the solution.

Step 5.5.2.3.1

First, use the positive value of the to find the first solution.

Step 5.5.2.3.2

Next, use the negative value of the to find the second solution.

Step 5.5.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

Step 5.6

The final solution is all the values that make true.

Step 6

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

Step 8

Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

Step 9

Step 9.1

Simplify each term.

Step 9.1.1

Raising to any positive power yields .

Step 9.1.2

Multiply by .

Step 9.2

Subtract from .

Step 10

is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

is a local maximum

Step 11

Step 11.1

Replace the variable with in the expression.

Step 11.2

Simplify the result.

Step 11.2.1

Simplify each term.

Step 11.2.1.1

Raising to any positive power yields .

Step 11.2.1.2

Raising to any positive power yields .

Step 11.2.1.3

Multiply by .

Step 11.2.2

Add and .

Step 11.2.3

The final answer is .

Step 12

Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

Step 13

Step 13.1

Simplify each term.

Step 13.1.1

Rewrite as .

Step 13.1.1.1

Use to rewrite as .

Step 13.1.1.2

Apply the power rule and multiply exponents, .

Step 13.1.1.3

Combine and .

Step 13.1.1.4

Cancel the common factor of .

Step 13.1.1.4.1

Cancel the common factor.

Step 13.1.1.4.2

Rewrite the expression.

Step 13.1.1.5

Evaluate the exponent.

Step 13.1.2

Multiply by .

Step 13.2

Subtract from .

Step 14

is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

is a local minimum

Step 15

Step 15.1

Replace the variable with in the expression.

Step 15.2

Simplify the result.

Step 15.2.1

Simplify each term.

Step 15.2.1.1

Rewrite as .

Step 15.2.1.1.1

Use to rewrite as .

Step 15.2.1.1.2

Apply the power rule and multiply exponents, .

Step 15.2.1.1.3

Combine and .

Step 15.2.1.1.4

Cancel the common factor of and .

Step 15.2.1.1.4.1

Factor out of .

Step 15.2.1.1.4.2

Cancel the common factors.

Step 15.2.1.1.4.2.1

Factor out of .

Step 15.2.1.1.4.2.2

Cancel the common factor.

Step 15.2.1.1.4.2.3

Rewrite the expression.

Step 15.2.1.1.4.2.4

Divide by .

Step 15.2.1.2

Raise to the power of .

Step 15.2.1.3

Rewrite as .

Step 15.2.1.3.1

Use to rewrite as .

Step 15.2.1.3.2

Apply the power rule and multiply exponents, .

Step 15.2.1.3.3

Combine and .

Step 15.2.1.3.4

Cancel the common factor of .

Step 15.2.1.3.4.1

Cancel the common factor.

Step 15.2.1.3.4.2

Rewrite the expression.

Step 15.2.1.3.5

Evaluate the exponent.

Step 15.2.1.4

Multiply by .

Step 15.2.2

Subtract from .

Step 15.2.3

The final answer is .

Step 16

Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

Step 17

Step 17.1

Simplify each term.

Step 17.1.1

Apply the product rule to .

Step 17.1.2

Raise to the power of .

Step 17.1.3

Multiply by .

Step 17.1.4

Rewrite as .

Step 17.1.4.1

Use to rewrite as .

Step 17.1.4.2

Apply the power rule and multiply exponents, .

Step 17.1.4.3

Combine and .

Step 17.1.4.4

Cancel the common factor of .

Step 17.1.4.4.1

Cancel the common factor.

Step 17.1.4.4.2

Rewrite the expression.

Step 17.1.4.5

Evaluate the exponent.

Step 17.1.5

Multiply by .

Step 17.2

Subtract from .

Step 18

is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

is a local minimum

Step 19

Step 19.1

Replace the variable with in the expression.

Step 19.2

Simplify the result.

Step 19.2.1

Simplify each term.

Step 19.2.1.1

Apply the product rule to .

Step 19.2.1.2

Raise to the power of .

Step 19.2.1.3

Multiply by .

Step 19.2.1.4

Rewrite as .

Step 19.2.1.4.1

Use to rewrite as .

Step 19.2.1.4.2

Apply the power rule and multiply exponents, .

Step 19.2.1.4.3

Combine and .

Step 19.2.1.4.4

Cancel the common factor of and .

Step 19.2.1.4.4.1

Factor out of .

Step 19.2.1.4.4.2

Cancel the common factors.

Step 19.2.1.4.4.2.1

Factor out of .

Step 19.2.1.4.4.2.2

Cancel the common factor.

Step 19.2.1.4.4.2.3

Rewrite the expression.

Step 19.2.1.4.4.2.4

Divide by .

Step 19.2.1.5

Raise to the power of .

Step 19.2.1.6

Apply the product rule to .

Step 19.2.1.7

Raise to the power of .

Step 19.2.1.8

Multiply by .

Step 19.2.1.9

Rewrite as .

Step 19.2.1.9.1

Use to rewrite as .

Step 19.2.1.9.2

Apply the power rule and multiply exponents, .

Step 19.2.1.9.3

Combine and .

Step 19.2.1.9.4

Cancel the common factor of .

Step 19.2.1.9.4.1

Cancel the common factor.

Step 19.2.1.9.4.2

Rewrite the expression.

Step 19.2.1.9.5

Evaluate the exponent.

Step 19.2.1.10

Multiply by .

Step 19.2.2

Subtract from .

Step 19.2.3

The final answer is .

Step 20

These are the local extrema for .

is a local maxima

is a local minima

is a local minima

Step 21