# Calculus Examples

Step 1

Step 1.1

Differentiate using the Power Rule which states that is where .

Step 1.2

The first derivative of with respect to is .

Step 2

Step 2.1

Set the first derivative equal to .

Step 2.2

Divide each term in by and simplify.

Step 2.2.1

Divide each term in by .

Step 2.2.2

Simplify the left side.

Step 2.2.2.1

Cancel the common factor of .

Step 2.2.2.1.1

Cancel the common factor.

Step 2.2.2.1.2

Divide by .

Step 2.2.3

Simplify the right side.

Step 2.2.3.1

Divide by .

Step 2.3

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

Step 2.4

Simplify .

Step 2.4.1

Rewrite as .

Step 2.4.2

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

Step 2.4.3

Plus or minus is .

Step 3

The values which make the derivative equal to are .

Step 4

After finding the point that makes the derivative equal to or undefined, the interval to check where is increasing and where it is decreasing is .

Step 5

Step 5.1

Replace the variable with in the expression.

Step 5.2

Simplify the result.

Step 5.2.1

Raise to the power of .

Step 5.2.2

Multiply by .

Step 5.2.3

The final answer is .

Step 5.3

At the derivative is . Since this is positive, the function is increasing on .

Increasing on since

Increasing on since

Step 6

Step 6.1

Replace the variable with in the expression.

Step 6.2

Simplify the result.

Step 6.2.1

One to any power is one.

Step 6.2.2

Multiply by .

Step 6.2.3

The final answer is .

Step 6.3

At the derivative is . Since this is positive, the function is increasing on .

Increasing on since

Increasing on since

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on:

Step 8