Calculus Examples

Find Where Increasing/Decreasing Using Derivatives f(x)=(x^2)/(x^2-4)
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 Quotient Rule which states that is where and .
Step 1.1.2
Differentiate.
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Step 1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2
Move to the left of .
Step 1.1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.6
Simplify the expression.
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Step 1.1.2.6.1
Add and .
Step 1.1.2.6.2
Multiply by .
Step 1.1.3
Raise to the power of .
Step 1.1.4
Use the power rule to combine exponents.
Step 1.1.5
Add and .
Step 1.1.6
Simplify.
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Step 1.1.6.1
Apply the distributive property.
Step 1.1.6.2
Apply the distributive property.
Step 1.1.6.3
Simplify the numerator.
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Step 1.1.6.3.1
Simplify each term.
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Step 1.1.6.3.1.1
Multiply by by adding the exponents.
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Step 1.1.6.3.1.1.1
Move .
Step 1.1.6.3.1.1.2
Multiply by .
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Step 1.1.6.3.1.1.2.1
Raise to the power of .
Step 1.1.6.3.1.1.2.2
Use the power rule to combine exponents.
Step 1.1.6.3.1.1.3
Add and .
Step 1.1.6.3.1.2
Multiply by .
Step 1.1.6.3.2
Combine the opposite terms in .
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Step 1.1.6.3.2.1
Subtract from .
Step 1.1.6.3.2.2
Add and .
Step 1.1.6.4
Move the negative in front of the fraction.
Step 1.1.6.5
Simplify the denominator.
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Step 1.1.6.5.1
Rewrite as .
Step 1.1.6.5.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.6.5.3
Apply the product rule to .
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
Set the numerator equal to zero.
Step 2.3
Divide each term in by and simplify.
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Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Cancel the common factor of .
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Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
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
Set the denominator in equal to to find where the expression is undefined.
Step 4.2
Solve for .
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Step 4.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.2.2
Set equal to and solve for .
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Step 4.2.2.1
Set equal to .
Step 4.2.2.2
Solve for .
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Step 4.2.2.2.1
Set the equal to .
Step 4.2.2.2.2
Subtract from both sides of the equation.
Step 4.2.3
Set equal to and solve for .
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Step 4.2.3.1
Set equal to .
Step 4.2.3.2
Solve for .
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Step 4.2.3.2.1
Set the equal to .
Step 4.2.3.2.2
Add to both sides of the equation.
Step 4.2.4
The final solution is all the values that make true.
Step 4.3
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
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
Multiply by .
Step 6.2.2
Simplify the denominator.
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Step 6.2.2.1
Add and .
Step 6.2.2.2
Subtract from .
Step 6.2.2.3
Raise to the power of .
Step 6.2.2.4
Raise to the power of .
Step 6.2.2.5
Multiply by .
Step 6.2.3
Move the negative in front of the fraction.
Step 6.2.4
The final answer is .
Step 6.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing 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
Multiply by .
Step 7.2.2
Simplify the denominator.
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Step 7.2.2.1
Add and .
Step 7.2.2.2
Subtract from .
Step 7.2.2.3
One to any power is one.
Step 7.2.2.4
Raise to the power of .
Step 7.2.2.5
Multiply by .
Step 7.2.3
Move the negative in front of the fraction.
Step 7.2.4
The final answer is .
Step 7.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing 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
Multiply by .
Step 8.2.2
Simplify the denominator.
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Step 8.2.2.1
Add and .
Step 8.2.2.2
Subtract from .
Step 8.2.2.3
Raise to the power of .
Step 8.2.2.4
Raise to the power of .
Step 8.2.3
Multiply by .
Step 8.2.4
The final answer is .
Step 8.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 9
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Step 9.1
Replace the variable with in the expression.
Step 9.2
Simplify the result.
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Step 9.2.1
Multiply by .
Step 9.2.2
Simplify the denominator.
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Step 9.2.2.1
Add and .
Step 9.2.2.2
Subtract from .
Step 9.2.2.3
Raise to the power of .
Step 9.2.2.4
One to any power is one.
Step 9.2.3
Multiply by .
Step 9.2.4
The final answer is .
Step 9.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 10
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 11