# Calculus Examples

Step 1

Step 1.1

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

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 1.3

Evaluate .

Step 1.3.1

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

Step 1.3.2

Differentiate using the Power Rule which states that is where .

Step 1.3.3

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

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

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 4.1.3

Evaluate .

Step 4.1.3.1

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

Step 4.1.3.2

Differentiate using the Power Rule which states that is where .

Step 4.1.3.3

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 and solve for .

Step 5.4.1

Set equal to .

Step 5.4.2

Solve for .

Step 5.4.2.1

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

Step 5.4.2.2

Simplify .

Step 5.4.2.2.1

Rewrite as .

Step 5.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

Step 5.4.2.2.3

Plus or minus is .

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

Divide each term in by and simplify.

Step 5.5.2.2.1

Divide each term in by .

Step 5.5.2.2.2

Simplify the left side.

Step 5.5.2.2.2.1

Cancel the common factor of .

Step 5.5.2.2.2.1.1

Cancel the common factor.

Step 5.5.2.2.2.1.2

Divide by .

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 9.1.3

Multiply by .

Step 9.2

Add and .

Step 10

Step 10.1

Split into separate intervals around the values that make the first derivative or undefined.

Step 10.2

Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.

Step 10.2.1

Replace the variable with in the expression.

Step 10.2.2

Simplify the result.

Step 10.2.2.1

Simplify each term.

Step 10.2.2.1.1

Raise to the power of .

Step 10.2.2.1.2

Multiply by .

Step 10.2.2.1.3

Raise to the power of .

Step 10.2.2.1.4

Multiply by .

Step 10.2.2.2

Subtract from .

Step 10.2.2.3

The final answer is .

Step 10.3

Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.

Step 10.3.1

Replace the variable with in the expression.

Step 10.3.2

Simplify the result.

Step 10.3.2.1

Simplify each term.

Step 10.3.2.1.1

Raise to the power of .

Step 10.3.2.1.2

Multiply by .

Step 10.3.2.1.3

Raise to the power of .

Step 10.3.2.1.4

Multiply by .

Step 10.3.2.2

Subtract from .

Step 10.3.2.3

The final answer is .

Step 10.4

Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.

Step 10.4.1

Replace the variable with in the expression.

Step 10.4.2

Simplify the result.

Step 10.4.2.1

Simplify each term.

Step 10.4.2.1.1

Raise to the power of .

Step 10.4.2.1.2

Multiply by .

Step 10.4.2.1.3

Raise to the power of .

Step 10.4.2.1.4

Multiply by .

Step 10.4.2.2

Subtract from .

Step 10.4.2.3

The final answer is .

Step 10.5

Since the first derivative did not change signs around , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.6

Since the first derivative changed signs from negative to positive around , then is a local minimum.

is a local minimum

is a local minimum

Step 11