# Calculus Examples

Find the second derivative.
Find the first 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 .
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 .
Divide each term by and simplify.
Divide each term in by .
Simplify the left side of the equation by cancelling the common factors.
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative one from the denominator of .
Simplify the expression.
Multiply by to get .
Rewrite as .
Divide by to get .
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.
Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
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.
Multiply by to get .
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.
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.
Multiply by to get .
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Replace the variable with in the expression.
Simplify the result.
Multiply by to get .
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
From to , the second derivative changes from increasing to decreasing. There is a valid inflection point at .
Inflection point at .
An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is .

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