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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
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
Evaluate .
Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.1.4.3
Multiply by .
Step 1.1.5
Differentiate using the Constant Rule.
Step 1.1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.5.2
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 the left side of the equation.
Step 2.2.1
Factor out of .
Step 2.2.1.1
Factor out of .
Step 2.2.1.2
Factor out of .
Step 2.2.1.3
Factor out of .
Step 2.2.1.4
Factor out of .
Step 2.2.1.5
Factor out of .
Step 2.2.2
Factor using the perfect square rule.
Step 2.2.2.1
Rewrite as .
Step 2.2.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.2.2.3
Rewrite the polynomial.
Step 2.2.2.4
Factor using the perfect square trinomial rule , where and .
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 .
Step 2.5
Set equal to and solve for .
Step 2.5.1
Set equal to .
Step 2.5.2
Solve for .
Step 2.5.2.1
Set the equal to .
Step 2.5.2.2
Subtract from 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
Raising to any positive power yields .
Step 4.1.2.1.2
Multiply by .
Step 4.1.2.1.3
Raising to any positive power yields .
Step 4.1.2.1.4
Multiply by .
Step 4.1.2.1.5
Raising to any positive power yields .
Step 4.1.2.1.6
Multiply by .
Step 4.1.2.2
Simplify by adding numbers.
Step 4.1.2.2.1
Add and .
Step 4.1.2.2.2
Add and .
Step 4.1.2.2.3
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
Multiply by .
Step 4.2.2.1.3
Raise to the power of .
Step 4.2.2.1.4
Multiply by .
Step 4.2.2.1.5
Raise to the power of .
Step 4.2.2.1.6
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
Add and .
Step 4.2.2.2.3
Add and .
Step 4.3
List all of the points.
Step 5