Calculus Examples

Find Where Increasing/Decreasing Using Derivatives x^2-x- natural log of x
Step 1
Write as a function.
Step 2
Find the first derivative.
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Step 2.1
Find the first derivative.
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Step 2.1.1
Differentiate.
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Step 2.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.1.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2
Evaluate .
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Step 2.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2.3
Multiply by .
Step 2.1.3
Evaluate .
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Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
The derivative of with respect to is .
Step 2.1.4
Reorder terms.
Step 2.2
The first derivative of with respect to is .
Step 3
Set the first derivative equal to then solve the equation .
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Step 3.1
Set the first derivative equal to .
Step 3.2
Find the LCD of the terms in the equation.
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Step 3.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 3.2.2
The LCM of one and any expression is the expression.
Step 3.3
Multiply each term in by to eliminate the fractions.
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Step 3.3.1
Multiply each term in by .
Step 3.3.2
Simplify the left side.
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Step 3.3.2.1
Simplify each term.
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Step 3.3.2.1.1
Multiply by by adding the exponents.
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Step 3.3.2.1.1.1
Move .
Step 3.3.2.1.1.2
Multiply by .
Step 3.3.2.1.2
Cancel the common factor of .
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Step 3.3.2.1.2.1
Move the leading negative in into the numerator.
Step 3.3.2.1.2.2
Cancel the common factor.
Step 3.3.2.1.2.3
Rewrite the expression.
Step 3.3.3
Simplify the right side.
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Step 3.3.3.1
Multiply by .
Step 3.4
Solve the equation.
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Step 3.4.1
Factor by grouping.
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Step 3.4.1.1
Reorder terms.
Step 3.4.1.2
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 3.4.1.2.1
Factor out of .
Step 3.4.1.2.2
Rewrite as plus
Step 3.4.1.2.3
Apply the distributive property.
Step 3.4.1.2.4
Multiply by .
Step 3.4.1.3
Factor out the greatest common factor from each group.
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Step 3.4.1.3.1
Group the first two terms and the last two terms.
Step 3.4.1.3.2
Factor out the greatest common factor (GCF) from each group.
Step 3.4.1.4
Factor the polynomial by factoring out the greatest common factor, .
Step 3.4.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.4.3
Set equal to and solve for .
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Step 3.4.3.1
Set equal to .
Step 3.4.3.2
Solve for .
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Step 3.4.3.2.1
Subtract from both sides of the equation.
Step 3.4.3.2.2
Divide each term in by and simplify.
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Step 3.4.3.2.2.1
Divide each term in by .
Step 3.4.3.2.2.2
Simplify the left side.
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Step 3.4.3.2.2.2.1
Cancel the common factor of .
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Step 3.4.3.2.2.2.1.1
Cancel the common factor.
Step 3.4.3.2.2.2.1.2
Divide by .
Step 3.4.3.2.2.3
Simplify the right side.
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Step 3.4.3.2.2.3.1
Move the negative in front of the fraction.
Step 3.4.4
Set equal to and solve for .
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Step 3.4.4.1
Set equal to .
Step 3.4.4.2
Add to both sides of the equation.
Step 3.4.5
The final solution is all the values that make true.
Step 4
The values which make the derivative equal to are .
Step 5
Set the denominator in equal to to find where the expression is undefined.
Step 6
Split into separate intervals around the values that make the derivative or undefined.
Step 7
Exclude the intervals that are not in the domain.
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 each term.
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Step 8.2.1.1
Multiply by .
Step 8.2.1.2
Divide by .
Step 8.2.1.3
Multiply by .
Step 8.2.2
Simplify by adding and subtracting.
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Step 8.2.2.1
Add and .
Step 8.2.2.2
Subtract from .
Step 8.2.3
The final answer is .
Step 9
Exclude the intervals that are not in the domain.
Step 10
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 10.1
Replace the variable with in the expression.
Step 10.2
Simplify the result.
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Step 10.2.1
Simplify each term.
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Step 10.2.1.1
Cancel the common factor of .
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Step 10.2.1.1.1
Cancel the common factor.
Step 10.2.1.1.2
Rewrite the expression.
Step 10.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 10.2.1.3
Multiply .
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Step 10.2.1.3.1
Multiply by .
Step 10.2.1.3.2
Multiply by .
Step 10.2.2
Simplify by subtracting numbers.
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Step 10.2.2.1
Subtract from .
Step 10.2.2.2
Subtract from .
Step 10.2.3
The final answer is .
Step 10.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 11
Exclude the intervals that are not in the domain.
Step 12
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 12.1
Replace the variable with in the expression.
Step 12.2
Simplify the result.
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Step 12.2.1
Multiply by .
Step 12.2.2
Find the common denominator.
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Step 12.2.2.1
Write as a fraction with denominator .
Step 12.2.2.2
Multiply by .
Step 12.2.2.3
Multiply by .
Step 12.2.2.4
Write as a fraction with denominator .
Step 12.2.2.5
Multiply by .
Step 12.2.2.6
Multiply by .
Step 12.2.3
Combine the numerators over the common denominator.
Step 12.2.4
Simplify each term.
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Step 12.2.4.1
Multiply by .
Step 12.2.4.2
Multiply by .
Step 12.2.5
Simplify by subtracting numbers.
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Step 12.2.5.1
Subtract from .
Step 12.2.5.2
Subtract from .
Step 12.2.6
The final answer is .
Step 12.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 13
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 14