# Find the Critical Points h(p)=(p-5)/(p^2+2)

Find the derivative.
Differentiate using the Quotient Rule which states that is where and .
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 .
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 .
Simplify.
Apply the distributive property.
Apply the distributive property.
Simplify the numerator.
Simplify each term.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Subtract from .
Factor out of .
Rewrite as .
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Move the negative in front of the fraction.
Set the derivative equal to .
Solve for .
Reorder terms.
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factors for are , which is multiplied by itself times.
occurs times.
The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.
Multiply each term by and simplify.
Multiply each term in by in order to remove all the denominators from the equation.
Simplify .
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Simplify .
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by by adding the exponents.
Use the power rule to combine exponents.
Move to the left of .
Multiply by .
Multiply by .
Solve the equation.
Factor out of .
Factor out of .
Factor out of .
Rewrite as .
Factor out of .
Factor out of .
Multiply each term in by
Multiply each term in by .
Simplify .
Apply the distributive property.
Simplify.
Multiply by .
Multiply by .
Apply the distributive property.
Simplify.
Multiply .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Multiply by .
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
The final answer is the combination of both solutions.
Substitute the values of which cause the derivative to be into the original function.
Evaluate.
Simplify the numerator.
Subtract from .
Simplify the denominator.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply and .
Expand the denominator using the FOIL method.
Simplify.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Apply the distributive property.
Move to the left of .
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Simplify each term.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Cancel the common factor of and .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify the numerator.
Subtract from .
Subtract from .
Simplify the denominator.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Multiply by .
Multiply .
Multiply by .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Subtract from .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Factor out of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply and .
Expand the denominator using the FOIL method.
Simplify.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Group and together.
Apply the distributive property.
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Simplify each term.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
Reduce the expression by cancelling the common factors.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Find the domain of the derivative.
Set the denominator in equal to to find where the expression is undefined.
Solve for .
Set the equal to .
Solve for .
Subtract from both sides of the equation.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Rewrite as .
Rewrite as .
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
The domain is all real numbers.
Interval Notation:
Set-Builder Notation:
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 h(p)=(p-5)/(p^2+2)