Calculus Examples
Step 1
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 .
Add and .
Step 2
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 .
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.
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 .
Add and .
The first derivative of with respect to is .
Step 5
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
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
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 .
Simplify by adding and subtracting.
Subtract from .
Add and .
The final answer is .
Step 11
These are the local extrema for .
is a local minima
Step 12