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Calculus Examples
Step 1
Step 1.1
By the Sum Rule, 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
Combine and .
Step 1.2.4
Multiply by .
Step 1.2.5
Combine and .
Step 1.2.6
Cancel the common factor of and .
Step 1.2.6.1
Factor out of .
Step 1.2.6.2
Cancel the common factors.
Step 1.2.6.2.1
Factor out of .
Step 1.2.6.2.2
Cancel the common factor.
Step 1.2.6.2.3
Rewrite the expression.
Step 1.2.6.2.4
Divide 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
Differentiate using the Constant Rule.
Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Add and .
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
By the Sum Rule, 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
Combine and .
Step 4.1.2.4
Multiply by .
Step 4.1.2.5
Combine and .
Step 4.1.2.6
Cancel the common factor of and .
Step 4.1.2.6.1
Factor out of .
Step 4.1.2.6.2
Cancel the common factors.
Step 4.1.2.6.2.1
Factor out of .
Step 4.1.2.6.2.2
Cancel the common factor.
Step 4.1.2.6.2.3
Rewrite the expression.
Step 4.1.2.6.2.4
Divide 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
Differentiate using the Constant Rule.
Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Add and .
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
Cancel the common factor of and .
Step 5.3.3.1.1
Factor out of .
Step 5.3.3.1.2
Cancel the common factors.
Step 5.3.3.1.2.1
Factor out of .
Step 5.3.3.1.2.2
Cancel the common factor.
Step 5.3.3.1.2.3
Rewrite the expression.
Step 5.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.5
Simplify .
Step 5.5.1
Rewrite as .
Step 5.5.2
Any root of is .
Step 5.5.3
Simplify the denominator.
Step 5.5.3.1
Rewrite as .
Step 5.5.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 5.6
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.6.1
First, use the positive value of the to find the first solution.
Step 5.6.2
Next, use the negative value of the to find the second solution.
Step 5.6.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Step 9.1
Factor out of .
Step 9.2
Cancel the common factor.
Step 9.3
Rewrite the expression.
Step 10
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 11
Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
Step 11.2.1
Simplify each term.
Step 11.2.1.1
Apply the product rule to .
Step 11.2.1.2
Combine.
Step 11.2.1.3
One to any power is one.
Step 11.2.1.4
Raise to the power of .
Step 11.2.1.5
Multiply by .
Step 11.2.1.6
Multiply by .
Step 11.2.1.7
Cancel the common factor of and .
Step 11.2.1.7.1
Factor out of .
Step 11.2.1.7.2
Cancel the common factors.
Step 11.2.1.7.2.1
Factor out of .
Step 11.2.1.7.2.2
Cancel the common factor.
Step 11.2.1.7.2.3
Rewrite the expression.
Step 11.2.1.8
Cancel the common factor of .
Step 11.2.1.8.1
Factor out of .
Step 11.2.1.8.2
Cancel the common factor.
Step 11.2.1.8.3
Rewrite the expression.
Step 11.2.2
Combine fractions.
Step 11.2.2.1
Combine the numerators over the common denominator.
Step 11.2.2.2
Add and .
Step 11.2.3
To write as a fraction with a common denominator, multiply by .
Step 11.2.4
Combine and .
Step 11.2.5
Combine the numerators over the common denominator.
Step 11.2.6
Simplify the numerator.
Step 11.2.6.1
Multiply by .
Step 11.2.6.2
Add and .
Step 11.2.7
Move the negative in front of the fraction.
Step 11.2.8
The final answer is .
Step 12
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 13
Step 13.1
Cancel the common factor of .
Step 13.1.1
Move the leading negative in into the numerator.
Step 13.1.2
Factor out of .
Step 13.1.3
Cancel the common factor.
Step 13.1.4
Rewrite the expression.
Step 13.2
Multiply by .
Step 14
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 15
Step 15.1
Replace the variable with in the expression.
Step 15.2
Simplify the result.
Step 15.2.1
Simplify each term.
Step 15.2.1.1
Use the power rule to distribute the exponent.
Step 15.2.1.1.1
Apply the product rule to .
Step 15.2.1.1.2
Apply the product rule to .
Step 15.2.1.2
Raise to the power of .
Step 15.2.1.3
One to any power is one.
Step 15.2.1.4
Raise to the power of .
Step 15.2.1.5
Cancel the common factor of .
Step 15.2.1.5.1
Move the leading negative in into the numerator.
Step 15.2.1.5.2
Cancel the common factor.
Step 15.2.1.5.3
Rewrite the expression.
Step 15.2.1.6
Combine and .
Step 15.2.1.7
Move the negative in front of the fraction.
Step 15.2.1.8
Cancel the common factor of .
Step 15.2.1.8.1
Move the leading negative in into the numerator.
Step 15.2.1.8.2
Factor out of .
Step 15.2.1.8.3
Cancel the common factor.
Step 15.2.1.8.4
Rewrite the expression.
Step 15.2.1.9
Multiply by .
Step 15.2.2
Combine fractions.
Step 15.2.2.1
Combine the numerators over the common denominator.
Step 15.2.2.2
Simplify the expression.
Step 15.2.2.2.1
Add and .
Step 15.2.2.2.2
Divide by .
Step 15.2.2.2.3
Add and .
Step 15.2.3
The final answer is .
Step 16
These are the local extrema for .
is a local minima
is a local maxima
Step 17