Calculus Examples

Find the Derivative - d/dx y=1/((x^2-1)(x^2+x+1))
Step 1
Rewrite as .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the Product Rule which states that is where and .
Step 4
Differentiate.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Differentiate using the Power Rule which states that is where .
Step 4.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.5
Add and .
Step 4.6
By the Sum Rule, the derivative of with respect to is .
Step 4.7
Differentiate using the Power Rule which states that is where .
Step 4.8
Since is constant with respect to , the derivative of with respect to is .
Step 4.9
Simplify the expression.
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Step 4.9.1
Add and .
Step 4.9.2
Move to the left of .
Step 5
Simplify.
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Step 5.1
Rewrite the expression using the negative exponent rule .
Step 5.2
Apply the product rule to .
Step 5.3
Apply the distributive property.
Step 5.4
Apply the distributive property.
Step 5.5
Combine terms.
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Step 5.5.1
Raise to the power of .
Step 5.5.2
Use the power rule to combine exponents.
Step 5.5.3
Add and .
Step 5.5.4
Raise to the power of .
Step 5.5.5
Raise to the power of .
Step 5.5.6
Use the power rule to combine exponents.
Step 5.5.7
Add and .
Step 5.5.8
Multiply by .
Step 5.6
Reorder the factors of .
Step 5.7
Simplify each term.
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Step 5.7.1
Expand using the FOIL Method.
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Step 5.7.1.1
Apply the distributive property.
Step 5.7.1.2
Apply the distributive property.
Step 5.7.1.3
Apply the distributive property.
Step 5.7.2
Simplify each term.
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Step 5.7.2.1
Rewrite using the commutative property of multiplication.
Step 5.7.2.2
Multiply by by adding the exponents.
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Step 5.7.2.2.1
Move .
Step 5.7.2.2.2
Multiply by .
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Step 5.7.2.2.2.1
Raise to the power of .
Step 5.7.2.2.2.2
Use the power rule to combine exponents.
Step 5.7.2.2.3
Add and .
Step 5.7.2.3
Multiply by .
Step 5.7.2.4
Multiply by .
Step 5.7.2.5
Multiply by .
Step 5.8
Combine the opposite terms in .
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Step 5.8.1
Add and .
Step 5.8.2
Add and .
Step 5.9
Add and .
Step 5.10
Add and .
Step 5.11
Apply the distributive property.
Step 5.12
Simplify.
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Step 5.12.1
Multiply by .
Step 5.12.2
Multiply by .
Step 5.12.3
Multiply by .
Step 5.13
Simplify the denominator.
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Step 5.13.1
Rewrite as .
Step 5.13.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 5.13.3
Apply the product rule to .
Step 5.14
Multiply by .
Step 5.15
Factor out of .
Step 5.16
Factor out of .
Step 5.17
Factor out of .
Step 5.18
Rewrite as .
Step 5.19
Factor out of .
Step 5.20
Rewrite as .
Step 5.21
Move the negative in front of the fraction.