# Calculus Examples

Differentiate using the Power Rule which states that is where .
Set the derivative equal to then solve for .
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 root of both sides of the to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is equal to .
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 .
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Raise to the power of to get .
Multiply by to get .
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Remove parentheses around .
One to any power is one.
Multiply by to get .
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:

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