# Calculus Examples

Step 1
Find the first derivative of the function.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
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 .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
Step 2
Find the second derivative of the function.
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 .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
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.
Find the first derivative.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
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 .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
Set the first derivative equal to .
Add to both sides of the equation.
Divide each term in by and simplify.
Divide each term in by .
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Step 6
Find the values where the derivative is undefined.
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 .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Multiply .
Combine and .
Multiply by .
Move the negative in front of the fraction.
Find the common denominator.
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.
Multiply by .
Multiply by .