# Calculus Examples

Find the second derivative.
Find the first derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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 .
Evaluate .
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 .
Find the second derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
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 .
Evaluate .
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 .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
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 .
Any root of is .
Multiply by .
Simplify.
Combine.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
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.
Apply the product rule to .
Simplify the numerator.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Apply the product rule to .
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by .
Evaluate the exponent.
Raise to the power of .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
The final answer is .
The point found by substituting in is . This point can be an inflection point.
Substitute in to find the value of .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Simplify the numerator.
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Reduce the expression by cancelling the common factors.
Cancel the common factor.
Divide by .
Evaluate the exponent.
Raise to the power of .
Reduce the expression by cancelling the common factors.
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Combine and .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
The final answer is .
The point found by substituting in is . This point can be an inflection point.
Determine the points that could be inflection points.
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.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
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.
Simplify each term.
Raising to any positive power yields .
Multiply by .
Subtract from .
The final answer is .
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing 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.
Simplify each term.
Raise to the power of .
Multiply by .
Subtract from .
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. The inflection points in this case are .