# Calculus Examples

Step 1
Find the first derivative.
Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Differentiate.
Step 1.4.1
Differentiate using the Power Rule which states that is where .
Step 1.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.5
Simplify.
Step 1.5.1
Step 1.5.2
Reorder terms.
Step 2
Graph each side of the equation. The solution is the x-value of the point of intersection.
Step 3
Split into separate intervals around the values that make the first derivative or undefined.
Step 4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Raising to any positive power yields .
Step 4.2.1.2
Multiply by .
Step 4.2.1.3
Raising to any positive power yields .
Step 4.2.1.4
Multiply by .
Step 4.2.2
Step 4.2.2.1
Step 4.2.2.2
Step 4.2.3
Step 5
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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
Raise to the power of .
Step 5.2.1.4
Multiply by .
Step 5.2.2
Step 5.2.2.1
Step 5.2.2.2
Step 5.2.3
Step 6
Since the first derivative changed signs from positive to negative around , then there is a turning point at .
Step 7
Find the y-coordinate of to find the turning point.
Step 7.1
Find to find the y-coordinate of .
Step 7.1.1
Replace the variable with in the expression.
Step 7.1.2
Simplify .
Step 7.1.2.1
Remove parentheses.
Step 7.1.2.2
Simplify each term.
Step 7.1.2.2.1
Raise to the power of .
Step 7.1.2.2.2
Multiply by .
Step 7.1.2.2.3
Raise to the power of .
Step 7.1.2.2.4
Multiply by .
Step 7.1.2.3