# Calculus Examples

Step 1

Step 1.1

Find the first derivative.

Step 1.1.1

Differentiate.

Step 1.1.1.1

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

Step 1.1.1.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2

Evaluate .

Step 1.1.2.1

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

Step 1.1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2.3

Multiply by .

Step 1.1.3

Differentiate using the Constant Rule.

Step 1.1.3.1

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

Step 1.1.3.2

Add and .

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

Subtract from both sides of the equation.

Step 2.3

Divide each term in by and simplify.

Step 2.3.1

Divide each term in by .

Step 2.3.2

Simplify the left side.

Step 2.3.2.1

Cancel the common factor of .

Step 2.3.2.1.1

Cancel the common factor.

Step 2.3.2.1.2

Divide by .

Step 2.3.3

Simplify the right side.

Step 2.3.3.1

Divide by .

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

Multiply by .

Step 5.2.2

Add and .

Step 5.2.3

The final answer is .

Step 5.3

At the derivative is . Since this is negative, the function is decreasing on .

Decreasing on since

Decreasing on since

Step 6

Step 6.1

Replace the variable with in the expression.

Step 6.2

Simplify the result.

Step 6.2.1

Multiply by .

Step 6.2.2

Add and .

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:

Decreasing on:

Step 8