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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 1.2
The first derivative of with respect to is .
Step 2
Step 2.1
Set the first derivative equal to .
Step 2.2
Factor out of .
Step 2.2.1
Factor out of .
Step 2.2.2
Factor out of .
Step 2.2.3
Factor out of .
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
Add to both sides of the equation.
Step 2.5.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.5.2.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 2.5.2.3.1
First, use the positive value of the to find the first solution.
Step 2.5.2.3.2
Next, use the negative value of the to find the second solution.
Step 2.5.2.3.3
The complete solution is the result of both the positive and negative portions of the solution.
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 4.1.2.2
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
Rewrite as .
Step 4.2.2.1.1.1
Use to rewrite as .
Step 4.2.2.1.1.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.1.3
Combine and .
Step 4.2.2.1.1.4
Cancel the common factor of and .
Step 4.2.2.1.1.4.1
Factor out of .
Step 4.2.2.1.1.4.2
Cancel the common factors.
Step 4.2.2.1.1.4.2.1
Factor out of .
Step 4.2.2.1.1.4.2.2
Cancel the common factor.
Step 4.2.2.1.1.4.2.3
Rewrite the expression.
Step 4.2.2.1.1.4.2.4
Divide by .
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
Rewrite as .
Step 4.2.2.1.4.1
Use to rewrite as .
Step 4.2.2.1.4.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.4.3
Combine and .
Step 4.2.2.1.4.4
Cancel the common factor of .
Step 4.2.2.1.4.4.1
Cancel the common factor.
Step 4.2.2.1.4.4.2
Rewrite the expression.
Step 4.2.2.1.4.5
Evaluate the exponent.
Step 4.2.2.1.5
Multiply by .
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
Apply the product rule to .
Step 4.3.2.1.2
Raise to the power of .
Step 4.3.2.1.3
Multiply by .
Step 4.3.2.1.4
Rewrite as .
Step 4.3.2.1.4.1
Use to rewrite as .
Step 4.3.2.1.4.2
Apply the power rule and multiply exponents, .
Step 4.3.2.1.4.3
Combine and .
Step 4.3.2.1.4.4
Cancel the common factor of and .
Step 4.3.2.1.4.4.1
Factor out of .
Step 4.3.2.1.4.4.2
Cancel the common factors.
Step 4.3.2.1.4.4.2.1
Factor out of .
Step 4.3.2.1.4.4.2.2
Cancel the common factor.
Step 4.3.2.1.4.4.2.3
Rewrite the expression.
Step 4.3.2.1.4.4.2.4
Divide by .
Step 4.3.2.1.5
Raise to the power of .
Step 4.3.2.1.6
Multiply by .
Step 4.3.2.1.7
Apply the product rule to .
Step 4.3.2.1.8
Raise to the power of .
Step 4.3.2.1.9
Multiply by .
Step 4.3.2.1.10
Rewrite as .
Step 4.3.2.1.10.1
Use to rewrite as .
Step 4.3.2.1.10.2
Apply the power rule and multiply exponents, .
Step 4.3.2.1.10.3
Combine and .
Step 4.3.2.1.10.4
Cancel the common factor of .
Step 4.3.2.1.10.4.1
Cancel the common factor.
Step 4.3.2.1.10.4.2
Rewrite the expression.
Step 4.3.2.1.10.5
Evaluate the exponent.
Step 4.3.2.1.11
Multiply by .
Step 4.3.2.2
Subtract from .
Step 4.4
List all of the points.
Step 5