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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
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
Evaluate .
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Evaluate .
Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.1.4.3
Multiply by .
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
Factor the left side of the equation.
Step 2.2.1
Factor out of .
Step 2.2.1.1
Factor out of .
Step 2.2.1.2
Factor out of .
Step 2.2.1.3
Factor out of .
Step 2.2.1.4
Factor out of .
Step 2.2.1.5
Factor out of .
Step 2.2.2
Factor.
Step 2.2.2.1
Factor using the AC method.
Step 2.2.2.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.2.2.1.2
Write the factored form using these integers.
Step 2.2.2.2
Remove unnecessary parentheses.
Step 2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.4
Set equal to and solve for .
Step 2.4.1
Set equal to .
Step 2.4.2
Add to both sides of the equation.
Step 2.5
Set equal to and solve for .
Step 2.5.1
Set equal to .
Step 2.5.2
Subtract from both sides of the equation.
Step 2.6
The final solution is all the values that make true.
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 5.2.1.3
Multiply by .
Step 5.2.2
Simplify by subtracting numbers.
Step 5.2.2.1
Subtract from .
Step 5.2.2.2
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
Use the power rule to distribute the exponent.
Step 6.2.1.1.1
Apply the product rule to .
Step 6.2.1.1.2
Apply the product rule to .
Step 6.2.1.2
Raise to the power of .
Step 6.2.1.3
Multiply by .
Step 6.2.1.4
One to any power is one.
Step 6.2.1.5
Raise to the power of .
Step 6.2.1.6
Cancel the common factor of .
Step 6.2.1.6.1
Factor out of .
Step 6.2.1.6.2
Factor out of .
Step 6.2.1.6.3
Cancel the common factor.
Step 6.2.1.6.4
Rewrite the expression.
Step 6.2.1.7
Combine and .
Step 6.2.1.8
Cancel the common factor of .
Step 6.2.1.8.1
Move the leading negative in into the numerator.
Step 6.2.1.8.2
Factor out of .
Step 6.2.1.8.3
Cancel the common factor.
Step 6.2.1.8.4
Rewrite the expression.
Step 6.2.1.9
Multiply by .
Step 6.2.2
Find the common denominator.
Step 6.2.2.1
Write as a fraction with denominator .
Step 6.2.2.2
Multiply by .
Step 6.2.2.3
Multiply by .
Step 6.2.2.4
Write as a fraction with denominator .
Step 6.2.2.5
Multiply by .
Step 6.2.2.6
Multiply by .
Step 6.2.3
Combine the numerators over the common denominator.
Step 6.2.4
Simplify each term.
Step 6.2.4.1
Multiply by .
Step 6.2.4.2
Multiply by .
Step 6.2.5
Simplify the expression.
Step 6.2.5.1
Subtract from .
Step 6.2.5.2
Subtract from .
Step 6.2.5.3
Move the negative in front of the fraction.
Step 6.2.6
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 7.2.1.3
Multiply by .
Step 7.2.2
Simplify by adding and subtracting.
Step 7.2.2.1
Add and .
Step 7.2.2.2
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