# Calculus Examples

Step 1
Find the second derivative.
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
Differentiate using the Power Rule which states that is where .
Step 1.1.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4
Step 1.2
Find the second derivative.
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
The second derivative of with respect to is .
Step 2
Set the second derivative equal to then solve the equation .
Step 2.1
Set the second 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 real numbers.
Step 3
Find the points where the second derivative is .
Step 3.1
Substitute in to find the value of .
Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
Step 3.1.2.1
Raising to any positive power yields .
Step 3.1.2.2
Subtract from .
Step 3.1.2.3
Step 3.2
The point found by substituting in is . This point can be an inflection point.
Step 4
Split into intervals around the points that could potentially be inflection points.
Step 5
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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
Step 5.3
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Step 6
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Raise to the power of .
Step 6.2.2
Multiply by .
Step 6.2.3