# Calculus Examples

Find the second derivative.
Find the first derivative.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Add and to get .
Find the second derivative.
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by to get .
The derivative of with respect to is .
Set the second derivative equal to then solve the equation .
Rewrite as a set of linear factors.
Divide each term by and simplify.
Divide each term in by .
Simplify the left side of the equation by cancelling the common factors.
Multiply by to get .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by to get .
Divide by to get .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
is equal to .
Find the points where the second derivative is .
Substitute in to find the value of .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Remove parentheses around .
Raising to any positive power yields .
Subtract from to get .
The final answer is .
The point found by substituting in is . This point can be an inflection point.
Split into intervals around the points that could potentially be inflection points.
There are no inflection points and the interval to test the concavity on is .
The graph is concave up because the second derivative is positive.
The graph is concave up