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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
Differentiate.
Step 1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2
Evaluate .
Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Multiply by .
Step 1.1.3
Evaluate .
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Differentiate using the Constant Rule.
Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Add and .
Step 1.2
The first derivative of with respect to is .
Step 2
Step 2.1
Set the first derivative equal to .
Step 2.2
Factor by grouping.
Step 2.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 2.2.1.1
Factor out of .
Step 2.2.1.2
Rewrite as plus
Step 2.2.1.3
Apply the distributive property.
Step 2.2.2
Factor out the greatest common factor from each group.
Step 2.2.2.1
Group the first two terms and the last two terms.
Step 2.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.4
Set equal to and solve for .
Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
Step 2.4.2.1
Subtract from both sides of the equation.
Step 2.4.2.2
Divide each term in by and simplify.
Step 2.4.2.2.1
Divide each term in by .
Step 2.4.2.2.2
Simplify the left side.
Step 2.4.2.2.2.1
Cancel the common factor of .
Step 2.4.2.2.2.1.1
Cancel the common factor.
Step 2.4.2.2.2.1.2
Divide by .
Step 2.4.2.2.3
Simplify the right side.
Step 2.4.2.2.3.1
Move the negative in front of the fraction.
Step 2.5
Set equal to and solve for .
Step 2.5.1
Set equal to .
Step 2.5.2
Add to both sides of the equation.
Step 2.6
The final solution is all the values that make true.
Step 3
Step 3.1
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 4
Step 4.1
Evaluate at .
Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
Step 4.1.2.1
Simplify each term.
Step 4.1.2.1.1
Use the power rule to distribute the exponent.
Step 4.1.2.1.1.1
Apply the product rule to .
Step 4.1.2.1.1.2
Apply the product rule to .
Step 4.1.2.1.2
Raise to the power of .
Step 4.1.2.1.3
Raise to the power of .
Step 4.1.2.1.4
Raise to the power of .
Step 4.1.2.1.5
Use the power rule to distribute the exponent.
Step 4.1.2.1.5.1
Apply the product rule to .
Step 4.1.2.1.5.2
Apply the product rule to .
Step 4.1.2.1.6
Raise to the power of .
Step 4.1.2.1.7
Multiply by .
Step 4.1.2.1.8
Raise to the power of .
Step 4.1.2.1.9
Raise to the power of .
Step 4.1.2.1.10
Multiply .
Step 4.1.2.1.10.1
Combine and .
Step 4.1.2.1.10.2
Multiply by .
Step 4.1.2.1.11
Move the negative in front of the fraction.
Step 4.1.2.1.12
Multiply .
Step 4.1.2.1.12.1
Multiply by .
Step 4.1.2.1.12.2
Combine and .
Step 4.1.2.1.12.3
Multiply by .
Step 4.1.2.2
Find the common denominator.
Step 4.1.2.2.1
Multiply by .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.2.3
Multiply by .
Step 4.1.2.2.4
Multiply by .
Step 4.1.2.2.5
Write as a fraction with denominator .
Step 4.1.2.2.6
Multiply by .
Step 4.1.2.2.7
Multiply by .
Step 4.1.2.2.8
Reorder the factors of .
Step 4.1.2.2.9
Multiply by .
Step 4.1.2.2.10
Multiply by .
Step 4.1.2.3
Combine the numerators over the common denominator.
Step 4.1.2.4
Simplify each term.
Step 4.1.2.4.1
Multiply by .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.4.3
Multiply by .
Step 4.1.2.5
Simplify by adding and subtracting.
Step 4.1.2.5.1
Subtract from .
Step 4.1.2.5.2
Add and .
Step 4.1.2.5.3
Add and .
Step 4.2
Evaluate at .
Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
Step 4.2.2.1
Simplify each term.
Step 4.2.2.1.1
Raise to the power of .
Step 4.2.2.1.2
Raise to the power of .
Step 4.2.2.1.3
Multiply by .
Step 4.2.2.1.4
Multiply by .
Step 4.2.2.2
Simplify by adding and subtracting.
Step 4.2.2.2.1
Subtract from .
Step 4.2.2.2.2
Subtract from .
Step 4.2.2.2.3
Add and .
Step 4.3
List all of the points.
Step 5