Calculus Examples

Find the first 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 .
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 .
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 .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Add to both sides of the equation.
Divide each term by and simplify.
Tap for more steps...
Divide each term in by .
Cancel the common factor of .
Tap for more steps...
Cancel the common factor.
Divide by .
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.
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
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 .
Cancel the common factor of .
Tap for more steps...
Factor out of .
Cancel the common factor.
Rewrite the expression.
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 and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Reorder the factors of .
Multiply by .
Combine fractions.
Tap for more steps...
Combine fractions with similar denominators.
Multiply by .
Simplify the numerator.
Tap for more steps...
Subtract from .
Add and .
The final answer is .
These are the local extrema for .
is a local minima
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