Calculus Examples

Find the Derivative - d/dy (y-3)/(y^2-3y+9)
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
Simplify the expression.
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Step 2.4.1
Add and .
Step 2.4.2
Multiply by .
Step 2.5
By the Sum Rule, the derivative of with respect to is .
Step 2.6
Differentiate using the Power Rule which states that is where .
Step 2.7
Since is constant with respect to , the derivative of with respect to is .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Multiply by .
Step 2.10
Since is constant with respect to , the derivative of with respect to is .
Step 2.11
Add and .
Step 3
Simplify.
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Step 3.1
Apply the distributive property.
Step 3.2
Simplify the numerator.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Multiply by .
Step 3.2.1.2
Expand using the FOIL Method.
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Step 3.2.1.2.1
Apply the distributive property.
Step 3.2.1.2.2
Apply the distributive property.
Step 3.2.1.2.3
Apply the distributive property.
Step 3.2.1.3
Simplify and combine like terms.
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Step 3.2.1.3.1
Simplify each term.
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Step 3.2.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 3.2.1.3.1.2
Multiply by by adding the exponents.
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Step 3.2.1.3.1.2.1
Move .
Step 3.2.1.3.1.2.2
Multiply by .
Step 3.2.1.3.1.3
Multiply by .
Step 3.2.1.3.1.4
Multiply by .
Step 3.2.1.3.1.5
Multiply by .
Step 3.2.1.3.1.6
Multiply by .
Step 3.2.1.3.2
Add and .
Step 3.2.2
Combine the opposite terms in .
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Step 3.2.2.1
Subtract from .
Step 3.2.2.2
Add and .
Step 3.2.3
Subtract from .
Step 3.2.4
Add and .
Step 3.3
Factor out of .
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Step 3.3.1
Factor out of .
Step 3.3.2
Factor out of .
Step 3.3.3
Factor out of .
Step 3.4
Factor out of .
Step 3.5
Rewrite as .
Step 3.6
Factor out of .
Step 3.7
Rewrite as .
Step 3.8
Move the negative in front of the fraction.