Calculus Examples

Find the second derivative.
Tap for more steps...
Find the first derivative.
Tap for more steps...
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Tap for more steps...
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 .
Tap for more steps...
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.
Tap for more steps...
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Tap for more steps...
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 .
Tap for more steps...
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 derivative of with respect to is .
Set the second derivative equal to then solve the equation .
Tap for more steps...
Add to both sides of the equation.
Divide each term by and simplify.
Tap for more steps...
Divide each term in by .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Cancel the common factor.
Divide by .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
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.
Tap for more steps...
Simplify the right side of the equation.
Tap for more steps...
Rewrite as .
Any root of is .
Multiply by .
Simplify.
Tap for more steps...
Combine.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
Multiply by .
The complete solution is the result of both the positive and negative portions of the solution.
Tap for more steps...
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 .
Tap for more steps...
Substitute in to find the value of .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
Apply the product rule to .
Simplify the numerator.
Tap for more steps...
Rewrite as .
Raise to the power of .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Simplify the denominator.
Tap for more steps...
Rewrite.
Remove parentheses around .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
Tap for more steps...
Write as a fraction with denominator .
Multiply and .
Apply the product rule to .
Rewrite as .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
Tap for more steps...
Write as a fraction with denominator .
Multiply 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 .
Tap for more steps...
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps...
Multiply by .
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 .
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
Apply the product rule to .
Raise to the power of .
Multiply by .
Apply the product rule to .
Simplify the numerator.
Tap for more steps...
Rewrite as .
Raise to the power of .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Simplify the denominator.
Tap for more steps...
Rewrite.
Remove parentheses around .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
Tap for more steps...
Write as a fraction with denominator .
Multiply and .
Apply the product rule to .
Raise to the power of .
Multiply by .
Apply the product rule to .
Rewrite as .
Raise to the power of .
Reduce the expression by cancelling the common factors.
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
Tap for more steps...
Write as a fraction with denominator .
Multiply 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 .
Tap for more steps...
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Tap for more steps...
Multiply by .
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.
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
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.
Tap for more steps...
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
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
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
Remove parentheses around .
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.
Tap for more steps...
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
Tap for more steps...
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
Tap for more steps...
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
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
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
Remove parentheses around .
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
Replace the variable with in the expression.
Simplify the result.
Tap for more steps...
Simplify each term.
Tap for more steps...
Remove parentheses around .
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
From to , the second derivative changes from increasing to decreasing. There is a valid inflection point at .
Inflection point at .
From to , the second derivative changes from decreasing to increasing. 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 points in this case are .
Enter YOUR Problem
Mathway requires javascript and a modern browser.