# Calculus Examples

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

Differentiate using the Power Rule which states that is where .

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

Add and to get .

Rewrite as a set of linear factors.

Divide each term by and simplify.

Divide each term in by .

Simplify the left side of the equation by cancelling the common factors.

Multiply by to get .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by to get .

Divide by to get .

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

Rewrite as .

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

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.

Raise to the power of to get .

Multiply by to get .

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.

Remove parentheses around .

One to any power is one.

Multiply by to get .

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: