Calculus Examples

Find Where Increasing/Decreasing Using Derivatives f(x)=x^(1/3)(x-4)
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
Differentiate using the Product Rule which states that is where and .
Step 1.1.2
Differentiate.
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Step 1.1.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.4
Simplify the expression.
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Step 1.1.2.4.1
Add and .
Step 1.1.2.4.2
Multiply by .
Step 1.1.2.5
Differentiate using the Power Rule which states that is where .
Step 1.1.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.4
Combine and .
Step 1.1.5
Combine the numerators over the common denominator.
Step 1.1.6
Simplify the numerator.
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Step 1.1.6.1
Multiply by .
Step 1.1.6.2
Subtract from .
Step 1.1.7
Move the negative in front of the fraction.
Step 1.1.8
Combine and .
Step 1.1.9
Move to the denominator using the negative exponent rule .
Step 1.1.10
Simplify.
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Step 1.1.10.1
Apply the distributive property.
Step 1.1.10.2
Combine terms.
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Step 1.1.10.2.1
Combine and .
Step 1.1.10.2.2
Move to the numerator using the negative exponent rule .
Step 1.1.10.2.3
Multiply by by adding the exponents.
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Step 1.1.10.2.3.1
Multiply by .
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Step 1.1.10.2.3.1.1
Raise to the power of .
Step 1.1.10.2.3.1.2
Use the power rule to combine exponents.
Step 1.1.10.2.3.2
Write as a fraction with a common denominator.
Step 1.1.10.2.3.3
Combine the numerators over the common denominator.
Step 1.1.10.2.3.4
Subtract from .
Step 1.1.10.2.4
Combine and .
Step 1.1.10.2.5
Move the negative in front of the fraction.
Step 1.1.10.2.6
To write as a fraction with a common denominator, multiply by .
Step 1.1.10.2.7
Combine and .
Step 1.1.10.2.8
Combine the numerators over the common denominator.
Step 1.1.10.2.9
Move to the left of .
Step 1.1.10.2.10
Add and .
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
Since has no factors besides and .
is a prime number
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
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 2.2.8
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
Multiply by by adding the exponents.
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Step 2.3.2.1.3.1
Move .
Step 2.3.2.1.3.2
Use the power rule to combine exponents.
Step 2.3.2.1.3.3
Combine the numerators over the common denominator.
Step 2.3.2.1.3.4
Add and .
Step 2.3.2.1.3.5
Divide by .
Step 2.3.2.1.4
Simplify .
Step 2.3.2.1.5
Cancel the common factor of .
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Step 2.3.2.1.5.1
Move the leading negative in into the numerator.
Step 2.3.2.1.5.2
Cancel the common factor.
Step 2.3.2.1.5.3
Rewrite the expression.
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
Add to 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
Divide by .
Step 3
The values which make the derivative equal to are .
Step 4
Find where the derivative is undefined.
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Step 4.1
Convert expressions with fractional exponents to radicals.
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Step 4.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 4.1.2
Apply the rule to rewrite the exponentiation as a radical.
Step 4.1.3
Anything raised to is the base itself.
Step 4.2
Set the denominator in equal to to find where the expression is undefined.
Step 4.3
Solve for .
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Step 4.3.1
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 4.3.2
Simplify each side of the equation.
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Step 4.3.2.1
Use to rewrite as .
Step 4.3.2.2
Simplify the left side.
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Step 4.3.2.2.1
Simplify .
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Step 4.3.2.2.1.1
Apply the product rule to .
Step 4.3.2.2.1.2
Raise to the power of .
Step 4.3.2.2.1.3
Multiply the exponents in .
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Step 4.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 4.3.2.2.1.3.2
Cancel the common factor of .
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Step 4.3.2.2.1.3.2.1
Cancel the common factor.
Step 4.3.2.2.1.3.2.2
Rewrite the expression.
Step 4.3.2.3
Simplify the right side.
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Step 4.3.2.3.1
Raising to any positive power yields .
Step 4.3.3
Solve for .
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Step 4.3.3.1
Divide each term in by and simplify.
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Step 4.3.3.1.1
Divide each term in by .
Step 4.3.3.1.2
Simplify the left side.
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Step 4.3.3.1.2.1
Cancel the common factor of .
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Step 4.3.3.1.2.1.1
Cancel the common factor.
Step 4.3.3.1.2.1.2
Divide by .
Step 4.3.3.1.3
Simplify the right side.
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Step 4.3.3.1.3.1
Divide by .
Step 4.3.3.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4.3.3.3
Simplify .
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Step 4.3.3.3.1
Rewrite as .
Step 4.3.3.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4.3.3.3.3
Plus or minus is .
Step 5
Split into separate intervals around the values that make the derivative or undefined.
Step 6
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
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Step 6.2.1
Simplify each term.
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Step 6.2.1.1
Simplify the numerator.
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Step 6.2.1.1.1
Rewrite as .
Step 6.2.1.1.2
Apply the power rule and multiply exponents, .
Step 6.2.1.1.3
Cancel the common factor of .
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Step 6.2.1.1.3.1
Cancel the common factor.
Step 6.2.1.1.3.2
Rewrite the expression.
Step 6.2.1.1.4
Evaluate the exponent.
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Move the negative in front of the fraction.
Step 6.2.1.4
Simplify the denominator.
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Step 6.2.1.4.1
Rewrite as .
Step 6.2.1.4.2
Apply the power rule and multiply exponents, .
Step 6.2.1.4.3
Cancel the common factor of .
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Step 6.2.1.4.3.1
Cancel the common factor.
Step 6.2.1.4.3.2
Rewrite the expression.
Step 6.2.1.4.4
Raise to the power of .
Step 6.2.1.5
Multiply by .
Step 6.2.2
Combine fractions.
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Step 6.2.2.1
Combine the numerators over the common denominator.
Step 6.2.2.2
Simplify the expression.
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Step 6.2.2.2.1
Subtract from .
Step 6.2.2.2.2
Move the negative in front of the fraction.
Step 6.2.3
The final answer is .
Step 6.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 7
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
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Step 7.2.1
Simplify each term.
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Step 7.2.1.1
Simplify the numerator.
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Step 7.2.1.1.1
Rewrite as .
Step 7.2.1.1.2
Rewrite as .
Step 7.2.1.1.3
Raise to the power of .
Step 7.2.1.1.4
Apply the power rule and multiply exponents, .
Step 7.2.1.1.5
Multiply by .
Step 7.2.1.1.6
Apply the power rule and multiply exponents, .
Step 7.2.1.1.7
Use the power rule to combine exponents.
Step 7.2.1.1.8
To write as a fraction with a common denominator, multiply by .
Step 7.2.1.1.9
Combine and .
Step 7.2.1.1.10
Combine the numerators over the common denominator.
Step 7.2.1.1.11
Simplify the numerator.
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Step 7.2.1.1.11.1
Multiply by .
Step 7.2.1.1.11.2
Subtract from .
Step 7.2.1.2
Simplify the denominator.
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Step 7.2.1.2.1
Apply the product rule to .
Step 7.2.1.2.2
One to any power is one.
Step 7.2.1.3
Combine and .
Step 7.2.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 7.2.1.5
Multiply .
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Step 7.2.1.5.1
Combine and .
Step 7.2.1.5.2
Rewrite as .
Step 7.2.1.5.3
Use the power rule to combine exponents.
Step 7.2.1.5.4
To write as a fraction with a common denominator, multiply by .
Step 7.2.1.5.5
Combine and .
Step 7.2.1.5.6
Combine the numerators over the common denominator.
Step 7.2.1.5.7
Simplify the numerator.
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Step 7.2.1.5.7.1
Multiply by .
Step 7.2.1.5.7.2
Add and .
Step 7.2.2
Combine the numerators over the common denominator.
Step 7.2.3
The final answer is .
Step 7.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 8
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 8.1
Replace the variable with in the expression.
Step 8.2
Simplify the result.
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Step 8.2.1
Simplify the numerator.
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Step 8.2.1.1
Rewrite as .
Step 8.2.1.2
Use the power rule to combine exponents.
Step 8.2.1.3
To write as a fraction with a common denominator, multiply by .
Step 8.2.1.4
Combine and .
Step 8.2.1.5
Combine the numerators over the common denominator.
Step 8.2.1.6
Simplify the numerator.
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Step 8.2.1.6.1
Multiply by .
Step 8.2.1.6.2
Add and .
Step 8.2.2
The final answer is .
Step 8.3
Simplify.
Step 8.4
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 9
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 10