Calculus Examples

Find the Derivative Using Quotient Rule - d/dx x/(x^2-9)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate.
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Step 2.1
Differentiate using the Power Rule which states that is where .
Step 2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
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
Multiply by .
Step 3.2.1.3
Rewrite using the commutative property of multiplication.
Step 3.2.1.4
Multiply by by adding the exponents.
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Step 3.2.1.4.1
Move .
Step 3.2.1.4.2
Multiply by .
Step 3.2.1.5
Multiply by .
Step 3.2.2
Subtract from .
Step 3.3
Simplify the denominator.
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Step 3.3.1
Rewrite as .
Step 3.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.3.3
Apply the product rule to .
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.