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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
Differentiate using the Power Rule which states that is where .
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
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
Evaluate .
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Evaluate .
Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Multiply by .
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
Differentiate using the Power Rule which states that is where .
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
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
Factor the left side of the equation.
Step 5.2.1
Factor out of .
Step 5.2.1.1
Factor out of .
Step 5.2.1.2
Factor out of .
Step 5.2.1.3
Factor out of .
Step 5.2.1.4
Factor out of .
Step 5.2.1.5
Factor out of .
Step 5.2.2
Factor.
Step 5.2.2.1
Factor by grouping.
Step 5.2.2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 5.2.2.1.1.1
Factor out of .
Step 5.2.2.1.1.2
Rewrite as plus
Step 5.2.2.1.1.3
Apply the distributive property.
Step 5.2.2.1.2
Factor out the greatest common factor from each group.
Step 5.2.2.1.2.1
Group the first two terms and the last two terms.
Step 5.2.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.2.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.2.2.2
Remove unnecessary parentheses.
Step 5.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4
Set equal to .
Step 5.5
Set equal to and solve for .
Step 5.5.1
Set equal to .
Step 5.5.2
Solve for .
Step 5.5.2.1
Subtract from both sides of the equation.
Step 5.5.2.2
Divide each term in by and simplify.
Step 5.5.2.2.1
Divide each term in by .
Step 5.5.2.2.2
Simplify the left side.
Step 5.5.2.2.2.1
Cancel the common factor of .
Step 5.5.2.2.2.1.1
Cancel the common factor.
Step 5.5.2.2.2.1.2
Divide by .
Step 5.5.2.2.3
Simplify the right side.
Step 5.5.2.2.3.1
Move the negative in front of the fraction.
Step 5.6
Set equal to and solve for .
Step 5.6.1
Set equal to .
Step 5.6.2
Subtract from both sides of the equation.
Step 5.7
The final solution is all the values that make true.
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
Simplify each term.
Step 9.1.1
Raising to any positive power yields .
Step 9.1.2
Multiply by .
Step 9.1.3
Multiply by .
Step 9.2
Simplify by adding numbers.
Step 9.2.1
Add and .
Step 9.2.2
Add and .
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
Raising to any positive power yields .
Step 11.2.1.2
Raising to any positive power yields .
Step 11.2.1.3
Multiply by .
Step 11.2.1.4
Raising to any positive power yields .
Step 11.2.1.5
Multiply by .
Step 11.2.2
Simplify by adding numbers.
Step 11.2.2.1
Add and .
Step 11.2.2.2
Add and .
Step 11.2.2.3
Add and .
Step 11.2.3
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
Simplify each term.
Step 13.1.1
Use the power rule to distribute the exponent.
Step 13.1.1.1
Apply the product rule to .
Step 13.1.1.2
Apply the product rule to .
Step 13.1.2
Raise to the power of .
Step 13.1.3
Multiply by .
Step 13.1.4
Raise to the power of .
Step 13.1.5
Raise to the power of .
Step 13.1.6
Cancel the common factor of .
Step 13.1.6.1
Factor out of .
Step 13.1.6.2
Cancel the common factor.
Step 13.1.6.3
Rewrite the expression.
Step 13.1.7
Multiply by .
Step 13.1.8
Cancel the common factor of .
Step 13.1.8.1
Move the leading negative in into the numerator.
Step 13.1.8.2
Factor out of .
Step 13.1.8.3
Cancel the common factor.
Step 13.1.8.4
Rewrite the expression.
Step 13.1.9
Multiply by .
Step 13.2
Simplify by adding and subtracting.
Step 13.2.1
Subtract from .
Step 13.2.2
Add and .
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
Multiply by .
Step 15.2.1.4
Raise to the power of .
Step 15.2.1.5
Raise to the power of .
Step 15.2.1.6
Use the power rule to distribute the exponent.
Step 15.2.1.6.1
Apply the product rule to .
Step 15.2.1.6.2
Apply the product rule to .
Step 15.2.1.7
Raise to the power of .
Step 15.2.1.8
Raise to the power of .
Step 15.2.1.9
Raise to the power of .
Step 15.2.1.10
Cancel the common factor of .
Step 15.2.1.10.1
Move the leading negative in into the numerator.
Step 15.2.1.10.2
Factor out of .
Step 15.2.1.10.3
Factor out of .
Step 15.2.1.10.4
Cancel the common factor.
Step 15.2.1.10.5
Rewrite the expression.
Step 15.2.1.11
Combine and .
Step 15.2.1.12
Multiply by .
Step 15.2.1.13
Move the negative in front of the fraction.
Step 15.2.1.14
Use the power rule to distribute the exponent.
Step 15.2.1.14.1
Apply the product rule to .
Step 15.2.1.14.2
Apply the product rule to .
Step 15.2.1.15
Raise to the power of .
Step 15.2.1.16
Multiply by .
Step 15.2.1.17
Raise to the power of .
Step 15.2.1.18
Raise to the power of .
Step 15.2.1.19
Multiply .
Step 15.2.1.19.1
Combine and .
Step 15.2.1.19.2
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
Add and .
Step 15.2.3
Find the common denominator.
Step 15.2.3.1
Write as a fraction with denominator .
Step 15.2.3.2
Multiply by .
Step 15.2.3.3
Multiply by .
Step 15.2.3.4
Multiply by .
Step 15.2.3.5
Multiply by .
Step 15.2.3.6
Multiply by .
Step 15.2.4
Combine the numerators over the common denominator.
Step 15.2.5
Simplify the expression.
Step 15.2.5.1
Multiply by .
Step 15.2.5.2
Add and .
Step 15.2.5.3
Add and .
Step 15.2.6
The final answer is .
Step 16
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 17
Step 17.1
Simplify each term.
Step 17.1.1
Raise to the power of .
Step 17.1.2
Multiply by .
Step 17.1.3
Multiply by .
Step 17.2
Simplify by adding and subtracting.
Step 17.2.1
Subtract from .
Step 17.2.2
Add and .
Step 18
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 19
Step 19.1
Replace the variable with in the expression.
Step 19.2
Simplify the result.
Step 19.2.1
Simplify each term.
Step 19.2.1.1
Raise to the power of .
Step 19.2.1.2
Raise to the power of .
Step 19.2.1.3
Multiply by .
Step 19.2.1.4
Raise to the power of .
Step 19.2.1.5
Multiply by .
Step 19.2.2
Simplify by adding and subtracting.
Step 19.2.2.1
Subtract from .
Step 19.2.2.2
Add and .
Step 19.2.2.3
Add and .
Step 19.2.3
The final answer is .
Step 20
These are the local extrema for .
is a local minima
is a local maxima
is a local minima
Step 21