Calculus Examples

Find the Derivative - d/dx y=(x^2-2x+1)/(x^2+5)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Differentiate using the Power Rule which states that is where .
Step 2.5
Multiply by .
Step 2.6
Since is constant with respect to , the derivative of with respect to is .
Step 2.7
Add and .
Step 2.8
By the Sum Rule, the derivative of with respect to is .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Since is constant with respect to , the derivative of with respect to is .
Step 2.11
Simplify the expression.
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Step 2.11.1
Add and .
Step 2.11.2
Multiply by .
Step 3
Simplify.
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Step 3.1
Apply the distributive property.
Step 3.2
Apply the distributive property.
Step 3.3
Simplify the numerator.
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Step 3.3.1
Simplify each term.
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Step 3.3.1.1
Expand using the FOIL Method.
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Step 3.3.1.1.1
Apply the distributive property.
Step 3.3.1.1.2
Apply the distributive property.
Step 3.3.1.1.3
Apply the distributive property.
Step 3.3.1.2
Simplify each term.
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Step 3.3.1.2.1
Rewrite using the commutative property of multiplication.
Step 3.3.1.2.2
Multiply by by adding the exponents.
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Step 3.3.1.2.2.1
Move .
Step 3.3.1.2.2.2
Multiply by .
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Step 3.3.1.2.2.2.1
Raise to the power of .
Step 3.3.1.2.2.2.2
Use the power rule to combine exponents.
Step 3.3.1.2.2.3
Add and .
Step 3.3.1.2.3
Move to the left of .
Step 3.3.1.2.4
Multiply by .
Step 3.3.1.2.5
Multiply by .
Step 3.3.1.3
Multiply by by adding the exponents.
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Step 3.3.1.3.1
Move .
Step 3.3.1.3.2
Multiply by .
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Step 3.3.1.3.2.1
Raise to the power of .
Step 3.3.1.3.2.2
Use the power rule to combine exponents.
Step 3.3.1.3.3
Add and .
Step 3.3.1.4
Multiply by by adding the exponents.
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Step 3.3.1.4.1
Move .
Step 3.3.1.4.2
Multiply by .
Step 3.3.1.5
Multiply by .
Step 3.3.1.6
Multiply by .
Step 3.3.2
Combine the opposite terms in .
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Step 3.3.2.1
Subtract from .
Step 3.3.2.2
Add and .
Step 3.3.3
Add and .
Step 3.3.4
Subtract from .
Step 3.4
Simplify the numerator.
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Step 3.4.1
Factor out of .
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Step 3.4.1.1
Factor out of .
Step 3.4.1.2
Factor out of .
Step 3.4.1.3
Factor out of .
Step 3.4.1.4
Factor out of .
Step 3.4.1.5
Factor out of .
Step 3.4.2
Factor using the AC method.
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Step 3.4.2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.4.2.2
Write the factored form using these integers.