Calculus Examples

Find the Derivative - d/dx y=(x+1)^2(x^2+1)^-3
Step 1
Rewrite as .
Step 2
Expand using the FOIL Method.
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Step 2.1
Apply the distributive property.
Step 2.2
Apply the distributive property.
Step 2.3
Apply the distributive property.
Step 3
Simplify and combine like terms.
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Step 3.1
Simplify each term.
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Step 3.1.1
Multiply by .
Step 3.1.2
Multiply by .
Step 3.1.3
Multiply by .
Step 3.1.4
Multiply by .
Step 3.2
Add and .
Step 4
Differentiate using the Product Rule which states that is where and .
Step 5
Differentiate using the chain rule, which states that is where and .
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Step 5.1
To apply the Chain Rule, set as .
Step 5.2
Differentiate using the Power Rule which states that is where .
Step 5.3
Replace all occurrences of with .
Step 6
Differentiate.
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Step 6.1
By the Sum Rule, the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Since is constant with respect to , the derivative of with respect to is .
Step 6.4
Simplify the expression.
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Step 6.4.1
Add and .
Step 6.4.2
Multiply by .
Step 6.5
By the Sum Rule, the derivative of with respect to is .
Step 6.6
Differentiate using the Power Rule which states that is where .
Step 6.7
Since is constant with respect to , the derivative of with respect to is .
Step 6.8
Differentiate using the Power Rule which states that is where .
Step 6.9
Multiply by .
Step 6.10
Since is constant with respect to , the derivative of with respect to is .
Step 6.11
Add and .
Step 7
Simplify.
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Step 7.1
Rewrite the expression using the negative exponent rule .
Step 7.2
Rewrite the expression using the negative exponent rule .
Step 7.3
Combine terms.
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Step 7.3.1
Combine and .
Step 7.3.2
Move the negative in front of the fraction.
Step 7.3.3
Combine and .
Step 7.3.4
Move to the left of .
Step 7.4
Reorder terms.
Step 7.5
Simplify each term.
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Step 7.5.1
Apply the distributive property.
Step 7.5.2
Simplify.
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Step 7.5.2.1
Multiply by .
Step 7.5.2.2
Multiply by .
Step 7.5.3
Multiply by .
Step 7.5.4
Simplify the numerator.
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Step 7.5.4.1
Factor by grouping.
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Step 7.5.4.1.1
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 7.5.4.1.1.1
Factor out of .
Step 7.5.4.1.1.2
Rewrite as plus
Step 7.5.4.1.1.3
Apply the distributive property.
Step 7.5.4.1.2
Factor out the greatest common factor from each group.
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Step 7.5.4.1.2.1
Group the first two terms and the last two terms.
Step 7.5.4.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 7.5.4.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 7.5.4.2
Combine exponents.
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Step 7.5.4.2.1
Factor out of .
Step 7.5.4.2.2
Rewrite as .
Step 7.5.4.2.3
Factor out of .
Step 7.5.4.2.4
Rewrite as .
Step 7.5.4.2.5
Raise to the power of .
Step 7.5.4.2.6
Raise to the power of .
Step 7.5.4.2.7
Use the power rule to combine exponents.
Step 7.5.4.2.8
Add and .
Step 7.5.4.2.9
Multiply by .
Step 7.5.5
Move the negative in front of the fraction.
Step 7.5.6
Multiply by .
Step 7.5.7
Factor out of .
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Step 7.5.7.1
Factor out of .
Step 7.5.7.2
Factor out of .
Step 7.5.7.3
Factor out of .
Step 7.6
To write as a fraction with a common denominator, multiply by .
Step 7.7
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 7.7.1
Multiply by .
Step 7.7.2
Multiply by by adding the exponents.
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Step 7.7.2.1
Multiply by .
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Step 7.7.2.1.1
Raise to the power of .
Step 7.7.2.1.2
Use the power rule to combine exponents.
Step 7.7.2.2
Add and .
Step 7.8
Combine the numerators over the common denominator.
Step 7.9
Simplify the numerator.
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Step 7.9.1
Factor out of .
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Step 7.9.1.1
Factor out of .
Step 7.9.1.2
Factor out of .
Step 7.9.2
Apply the distributive property.
Step 7.9.3
Multiply by .
Step 7.9.4
Apply the distributive property.
Step 7.9.5
Multiply by by adding the exponents.
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Step 7.9.5.1
Move .
Step 7.9.5.2
Multiply by .
Step 7.9.6
Add and .
Step 7.10
Factor out of .
Step 7.11
Factor out of .
Step 7.12
Factor out of .
Step 7.13
Rewrite as .
Step 7.14
Factor out of .
Step 7.15
Rewrite as .
Step 7.16
Move the negative in front of the fraction.
Step 7.17
Reorder factors in .