Calculus Examples

Find the Critical Points f(x)=x^(20/21)-x^(41/21)
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Evaluate .
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Step 1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.1.2.3
Combine and .
Step 1.1.2.4
Combine the numerators over the common denominator.
Step 1.1.2.5
Simplify the numerator.
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Step 1.1.2.5.1
Multiply by .
Step 1.1.2.5.2
Subtract from .
Step 1.1.2.6
Move the negative in front of the fraction.
Step 1.1.3
Evaluate .
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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
To write as a fraction with a common denominator, multiply by .
Step 1.1.3.4
Combine and .
Step 1.1.3.5
Combine the numerators over the common denominator.
Step 1.1.3.6
Simplify the numerator.
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Step 1.1.3.6.1
Multiply by .
Step 1.1.3.6.2
Subtract from .
Step 1.1.3.7
Combine and .
Step 1.1.4
Simplify.
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Step 1.1.4.1
Rewrite the expression using the negative exponent rule .
Step 1.1.4.2
Multiply by .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Find the LCD of the terms in the equation.
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Step 2.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 2.2.2
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 2.2.3
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.
Step 2.2.4
has factors of and .
Step 2.2.5
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 2.2.6
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
Step 2.2.7
Multiply by .
Step 2.2.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 2.2.9
The LCM for is the numeric part multiplied by the variable part.
Step 2.3
Multiply each term in by to eliminate the fractions.
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Step 2.3.1
Multiply each term in by .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Simplify each term.
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Step 2.3.2.1.1
Rewrite using the commutative property of multiplication.
Step 2.3.2.1.2
Cancel the common factor of .
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Step 2.3.2.1.2.1
Cancel the common factor.
Step 2.3.2.1.2.2
Rewrite the expression.
Step 2.3.2.1.3
Cancel the common factor of .
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Step 2.3.2.1.3.1
Cancel the common factor.
Step 2.3.2.1.3.2
Rewrite the expression.
Step 2.3.2.1.4
Cancel the common factor of .
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Step 2.3.2.1.4.1
Move the leading negative in into the numerator.
Step 2.3.2.1.4.2
Factor out of .
Step 2.3.2.1.4.3
Cancel the common factor.
Step 2.3.2.1.4.4
Rewrite the expression.
Step 2.3.2.1.5
Multiply by by adding the exponents.
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Step 2.3.2.1.5.1
Move .
Step 2.3.2.1.5.2
Use the power rule to combine exponents.
Step 2.3.2.1.5.3
Combine the numerators over the common denominator.
Step 2.3.2.1.5.4
Add and .
Step 2.3.2.1.5.5
Divide by .
Step 2.3.2.1.6
Simplify .
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
Multiply .
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Step 2.3.3.1.1
Multiply by .
Step 2.3.3.1.2
Multiply by .
Step 2.4
Solve the equation.
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Step 2.4.1
Subtract from both sides of the equation.
Step 2.4.2
Divide each term in by and simplify.
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Step 2.4.2.1
Divide each term in by .
Step 2.4.2.2
Simplify the left side.
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Step 2.4.2.2.1
Cancel the common factor of .
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Step 2.4.2.2.1.1
Cancel the common factor.
Step 2.4.2.2.1.2
Divide by .
Step 2.4.2.3
Simplify the right side.
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Step 2.4.2.3.1
Dividing two negative values results in a positive value.
Step 3
Find the values where the derivative is undefined.
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Step 3.1
Convert expressions with fractional exponents to radicals.
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Step 3.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 3.1.2
Apply the rule to rewrite the exponentiation as a radical.
Step 3.1.3
Anything raised to is the base itself.
Step 3.2
Set the denominator in equal to to find where the expression is undefined.
Step 3.3
Solve for .
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Step 3.3.1
To remove the radical on the left side of the equation, raise both sides of the equation to the power of .
Step 3.3.2
Simplify each side of the equation.
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Step 3.3.2.1
Use to rewrite as .
Step 3.3.2.2
Simplify the left side.
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Step 3.3.2.2.1
Simplify .
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Step 3.3.2.2.1.1
Apply the product rule to .
Step 3.3.2.2.1.2
Raise to the power of .
Step 3.3.2.2.1.3
Multiply the exponents in .
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Step 3.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 3.3.2.2.1.3.2
Cancel the common factor of .
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Step 3.3.2.2.1.3.2.1
Cancel the common factor.
Step 3.3.2.2.1.3.2.2
Rewrite the expression.
Step 3.3.2.2.1.4
Simplify.
Step 3.3.2.3
Simplify the right side.
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Step 3.3.2.3.1
Raising to any positive power yields .
Step 3.3.3
Divide each term in by and simplify.
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Step 3.3.3.1
Divide each term in by .
Step 3.3.3.2
Simplify the left side.
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Step 3.3.3.2.1
Cancel the common factor of .
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Step 3.3.3.2.1.1
Cancel the common factor.
Step 3.3.3.2.1.2
Divide by .
Step 3.3.3.3
Simplify the right side.
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Step 3.3.3.3.1
Divide by .
Step 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
Simplify each term.
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Step 4.1.2.1.1
Apply the product rule to .
Step 4.1.2.1.2
Apply the product rule to .
Step 4.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 4.1.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 4.1.2.3.1
Multiply by .
Step 4.1.2.3.2
Multiply by by adding the exponents.
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Step 4.1.2.3.2.1
Use the power rule to combine exponents.
Step 4.1.2.3.2.2
Combine the numerators over the common denominator.
Step 4.1.2.3.2.3
Add and .
Step 4.1.2.4
Reduce the expression by cancelling the common factors.
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Step 4.1.2.4.1
Combine the numerators over the common denominator.
Step 4.1.2.4.2
Cancel the common factor of .
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Step 4.1.2.4.2.1
Cancel the common factor.
Step 4.1.2.4.2.2
Rewrite the expression.
Step 4.1.2.5
Simplify the numerator.
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Step 4.1.2.5.1
Evaluate the exponent.
Step 4.1.2.5.2
Move to the left of .
Step 4.2
Evaluate at .
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Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
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Step 4.2.2.1
Simplify each term.
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Step 4.2.2.1.1
Rewrite as .
Step 4.2.2.1.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.3
Cancel the common factor of .
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Step 4.2.2.1.3.1
Cancel the common factor.
Step 4.2.2.1.3.2
Rewrite the expression.
Step 4.2.2.1.4
Raising to any positive power yields .
Step 4.2.2.1.5
Rewrite as .
Step 4.2.2.1.6
Apply the power rule and multiply exponents, .
Step 4.2.2.1.7
Cancel the common factor of .
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Step 4.2.2.1.7.1
Cancel the common factor.
Step 4.2.2.1.7.2
Rewrite the expression.
Step 4.2.2.1.8
Raising to any positive power yields .
Step 4.2.2.1.9
Multiply by .
Step 4.2.2.2
Add and .
Step 4.3
List all of the points.
Step 5