# Calculus Examples

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

Differentiate using the Power Rule which states that is where .

Evaluate .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

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

Add and .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Simplify .

Divide by .

Multiply by .

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 .

Replace the variable with in the expression.

Simplify the result.

Multiply by .

Add and .

The final answer is .

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

Decreasing on since

Decreasing on since

Replace the variable with in the expression.

Simplify the result.

Multiply by .

Add and .

The final answer is .

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

Increasing on since

Increasing on since

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

Increasing on:

Decreasing on: