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Calculus Examples
Step 1
Step 1.1
Differentiate.
Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Evaluate .
Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Simplify.
Step 1.4.1
Subtract from .
Step 1.4.2
Reorder terms.
Step 2
Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Differentiate using the Constant Rule.
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Add and .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Step 4.1
Find the first derivative.
Step 4.1.1
Differentiate.
Step 4.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2
Evaluate .
Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.1.3
Evaluate .
Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Multiply by .
Step 4.1.4
Simplify.
Step 4.1.4.1
Subtract from .
Step 4.1.4.2
Reorder terms.
Step 4.2
The first derivative of with respect to is .
Step 5
Step 5.1
Set the first derivative equal to .
Step 5.2
Add to both sides of the equation.
Step 5.3
Divide each term in by and simplify.
Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
Step 5.3.2.1
Cancel the common factor of .
Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Divide by .
Step 5.3.3
Simplify the right side.
Step 5.3.3.1
Move the negative in front of the fraction.
Step 6
Step 6.1
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 maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 10
Step 10.1
Replace the variable with in the expression.
Step 10.2
Simplify the result.
Step 10.2.1
Simplify each term.
Step 10.2.1.1
Multiply .
Step 10.2.1.1.1
Multiply by .
Step 10.2.1.1.2
Multiply by .
Step 10.2.1.2
Use the power rule to distribute the exponent.
Step 10.2.1.2.1
Apply the product rule to .
Step 10.2.1.2.2
Apply the product rule to .
Step 10.2.1.3
Multiply by by adding the exponents.
Step 10.2.1.3.1
Move .
Step 10.2.1.3.2
Multiply by .
Step 10.2.1.3.2.1
Raise to the power of .
Step 10.2.1.3.2.2
Use the power rule to combine exponents.
Step 10.2.1.3.3
Add and .
Step 10.2.1.4
Raise to the power of .
Step 10.2.1.5
One to any power is one.
Step 10.2.1.6
Raise to the power of .
Step 10.2.2
Find the common denominator.
Step 10.2.2.1
Write as a fraction with denominator .
Step 10.2.2.2
Multiply by .
Step 10.2.2.3
Multiply by .
Step 10.2.2.4
Multiply by .
Step 10.2.2.5
Multiply by .
Step 10.2.2.6
Multiply by .
Step 10.2.3
Combine the numerators over the common denominator.
Step 10.2.4
Simplify the expression.
Step 10.2.4.1
Multiply by .
Step 10.2.4.2
Add and .
Step 10.2.4.3
Subtract from .
Step 10.2.5
The final answer is .
Step 11
These are the local extrema for .
is a local maxima
Step 12