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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
Add to 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 2.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.5
Any root of is .
Step 2.6
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.6.1
First, use the positive value of the to find the first solution.
Step 2.6.2
Next, use the negative value of the to find the second solution.
Step 2.6.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 3
The values which make the derivative equal to are .
Step 4
Split into separate intervals around the values that make the derivative or undefined.
Step 5
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Multiply by .
Step 5.2.2
Subtract from .
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
Simplify each term.
Step 6.2.1.1
Raising to any positive power yields .
Step 6.2.1.2
Multiply by .
Step 6.2.2
Subtract from .
Step 6.2.3
The final answer is .
Step 6.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 7
Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
Step 7.2.1
Simplify each term.
Step 7.2.1.1
Raise to the power of .
Step 7.2.1.2
Multiply by .
Step 7.2.2
Subtract from .
Step 7.2.3
The final answer is .
Step 7.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 8
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 9