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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
Combine and .
Step 1.1.2.4
Combine and .
Step 1.1.2.5
Cancel the common factor of .
Step 1.1.2.5.1
Cancel the common factor.
Step 1.1.2.5.2
Divide 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 1.2
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 2.2.2
Rewrite as .
Step 2.2.3
Factor.
Step 2.2.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.2.3.2
Remove unnecessary parentheses.
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
Subtract from both sides of the equation.
Step 2.6
Set equal to and solve for .
Step 2.6.1
Set equal to .
Step 2.6.2
Add to both sides of the equation.
Step 2.7
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 4.1.2.2
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
Cancel the common factor of .
Step 4.2.2.1.2.1
Factor out of .
Step 4.2.2.1.2.2
Cancel the common factor.
Step 4.2.2.1.2.3
Rewrite the expression.
Step 4.2.2.1.3
Multiply by by adding the exponents.
Step 4.2.2.1.3.1
Multiply by .
Step 4.2.2.1.3.1.1
Raise to the power of .
Step 4.2.2.1.3.1.2
Use the power rule to combine exponents.
Step 4.2.2.1.3.2
Add and .
Step 4.2.2.1.4
Raise to the power of .
Step 4.2.2.2
Subtract from .
Step 4.3
Evaluate at .
Step 4.3.1
Substitute for .
Step 4.3.2
Simplify.
Step 4.3.2.1
Simplify each term.
Step 4.3.2.1.1
Raise to the power of .
Step 4.3.2.1.2
Cancel the common factor of .
Step 4.3.2.1.2.1
Factor out of .
Step 4.3.2.1.2.2
Cancel the common factor.
Step 4.3.2.1.2.3
Rewrite the expression.
Step 4.3.2.1.3
Raise to the power of .
Step 4.3.2.1.4
Multiply by .
Step 4.3.2.2
Subtract from .
Step 4.4
List all of the points.
Step 5