# 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 .

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 .

The second derivative of with respect to is .

Set the second derivative equal to then solve the equation .

Set the second derivative equal to .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

Take the root of both sides of the 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.

The absolute value is the distance between a number and zero. The distance between and is .

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.

Raising to any positive power yields .

Subtract from .

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.

Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.

Replace the variable with in the expression.

Simplify the result.

Raise to the power of .

Multiply by .

The final answer is .

At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .

Increasing on since

Increasing on since

Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.

Replace the variable with in the expression.

Simplify the result.

Raise to the power of .

Multiply by .

The final answer is .

At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .

Increasing on since

Increasing on since

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. There are no points on the graph that satisfy these requirements.

No Inflection Points

No Inflection Points

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

Set-Builder Notation:

The graph is concave up because the second derivative is positive.

The graph is concave up