Calculus Examples

Find the second derivative.
Tap for more steps...
Find the first derivative.
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 to get .
Find the second derivative.
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 to get .
The derivative of with respect to is .
Divide each term by and simplify.
Tap for more steps...
Divide each term in by .
Simplify the left side of the equation by cancelling the common factors.
Tap for more steps...
Reduce the expression by cancelling the common factors.
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative one from the denominator of .
Simplify the expression.
Tap for more steps...
Multiply by to get .
Rewrite as .
Divide by to get .
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...
Remove parentheses around .
Raising to any positive power yields .
Multiply by 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.
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...
Multiply by to get .
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...
Multiply by to get .
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...
Multiply by to get .
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
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 .
Enter YOUR Problem

Enter the email address associated with your Mathway account below and we'll send you a link to reset your password.

Please enter an email address
Please enter a valid email address
The email address you entered was not found in our system
The email address you entered is associated with a Facebook user
We're sorry, we were unable to process your request at this time

Mathway Premium

Step-by-step work + explanations
  •    Step-by-step work
  •    Detailed explanations
  •    No advertisements
  •    Access anywhere
Access the steps on both the Mathway website and mobile apps
$--.--/month
$--.--/year (--%)

Mathway Premium

Visa and MasterCard security codes are located on the back of card and are typically a separate group of 3 digits to the right of the signature strip.

American Express security codes are 4 digits located on the front of the card and usually towards the right.
This option is required to subscribe.
Go Back

Step-by-step upgrade complete!

Mathway requires javascript and a modern browser.
  [ x 2     1 2     π     x d x   ]