# Find the Critical Points (x^2-1)^3 Find the derivative.
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
Multiply by .
Reorder the factors of .
Set the derivative equal to .
Solve for .
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to .
Set the next factor equal to and solve.
Set the next factor equal to .
Factor the left side of the equation.
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Apply the product rule to .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Set the equal to .
Subtract from both sides of the equation.
Set the next factor equal to and solve.
Set the next factor equal to .
Set the equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
The final solution is all the values that make true.
Substitute the values of which cause the derivative to be into the original function.
Evaluate.
Raising to any positive power yields .
Subtract from .
Raise to the power of .
Raise to the power of .
Subtract from .
Raising to any positive power yields .
One to any power is one.
Subtract from .
Raising to any positive power yields .
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.
Interval Notation:
Set-Builder Notation:
Since there are no values of where the derivative is undefined, there are no additional critical points.
Find the Critical Points (x^2-1)^3     