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