Calculus Examples

Step 1
Find the first derivative of the function.
Tap for more steps...
Differentiate.
Tap for more steps...
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
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 .
Differentiate using the Constant Rule.
Tap for more steps...
Since is constant with respect to , the derivative of with respect to is .
Add and .
Step 2
Find the second derivative of the function.
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 .
Differentiate using the Constant Rule.
Tap for more steps...
Since is constant with respect to , the derivative of with respect to is .
Add and .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Find the first derivative.
Tap for more steps...
Find the first derivative.
Tap for more steps...
Differentiate.
Tap for more steps...
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
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 .
Differentiate using the Constant Rule.
Tap for more steps...
Since is constant with respect to , the derivative of with respect to is .
Add and .
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
Tap for more steps...
Set the first derivative equal to .
Add to both sides of the equation.
Divide each term in by and simplify.
Tap for more steps...
Divide each term in by .
Simplify the left side.
Tap for more steps...
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Divide by .
Step 6
Find the values where the derivative is undefined.
Tap for more steps...
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.
Step 7
Critical points to evaluate.
Step 8
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 9
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 10
Find the y-value when .
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 .
Raise to the power of .
Multiply .
Tap for more steps...
Combine and .
Multiply by .
Move the negative in front of the fraction.
Find the common denominator.
Tap for more steps...
Multiply by .
Multiply by .
Write as a fraction with denominator .
Multiply by .
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Simplify each term.
Tap for more steps...
Multiply by .
Multiply by .
Simplify by adding and subtracting.
Tap for more steps...
Subtract from .
Add and .
The final answer is .
Step 11
These are the local extrema for .
is a local minima
Step 12
Enter YOUR Problem
Mathway requires javascript and a modern browser.
Cookies & Privacy
This website uses cookies to ensure you get the best experience on our website.
More Information