Calculus Examples

Find the Derivative - d/dy (13y^2)/((7x+13y)^2)
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Differentiate using the Power Rule.
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Step 3.1
Multiply the exponents in .
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Step 3.1.1
Apply the power rule and multiply exponents, .
Step 3.1.2
Multiply by .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Move to the left of .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
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Step 5.1
Multiply by .
Step 5.2
By the Sum Rule, the derivative of with respect to is .
Step 5.3
Since is constant with respect to , the derivative of with respect to is .
Step 5.4
Add and .
Step 5.5
Since is constant with respect to , the derivative of with respect to is .
Step 5.6
Simplify the expression.
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Step 5.6.1
Move to the left of .
Step 5.6.2
Multiply by .
Step 5.7
Differentiate using the Power Rule which states that is where .
Step 5.8
Combine fractions.
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Step 5.8.1
Multiply by .
Step 5.8.2
Combine and .
Step 6
Simplify.
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Step 6.1
Apply the distributive property.
Step 6.2
Apply the distributive property.
Step 6.3
Simplify the numerator.
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Step 6.3.1
Simplify each term.
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Step 6.3.1.1
Rewrite as .
Step 6.3.1.2
Expand using the FOIL Method.
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Step 6.3.1.2.1
Apply the distributive property.
Step 6.3.1.2.2
Apply the distributive property.
Step 6.3.1.2.3
Apply the distributive property.
Step 6.3.1.3
Simplify and combine like terms.
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Step 6.3.1.3.1
Simplify each term.
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Step 6.3.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 6.3.1.3.1.2
Multiply by by adding the exponents.
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Step 6.3.1.3.1.2.1
Move .
Step 6.3.1.3.1.2.2
Multiply by .
Step 6.3.1.3.1.3
Multiply by .
Step 6.3.1.3.1.4
Rewrite using the commutative property of multiplication.
Step 6.3.1.3.1.5
Multiply by .
Step 6.3.1.3.1.6
Rewrite using the commutative property of multiplication.
Step 6.3.1.3.1.7
Multiply by .
Step 6.3.1.3.1.8
Rewrite using the commutative property of multiplication.
Step 6.3.1.3.1.9
Multiply by by adding the exponents.
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Step 6.3.1.3.1.9.1
Move .
Step 6.3.1.3.1.9.2
Multiply by .
Step 6.3.1.3.1.10
Multiply by .
Step 6.3.1.3.2
Add and .
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Step 6.3.1.3.2.1
Move .
Step 6.3.1.3.2.2
Add and .
Step 6.3.1.4
Apply the distributive property.
Step 6.3.1.5
Simplify.
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Step 6.3.1.5.1
Multiply by .
Step 6.3.1.5.2
Multiply by .
Step 6.3.1.5.3
Multiply by .
Step 6.3.1.6
Apply the distributive property.
Step 6.3.1.7
Simplify.
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Step 6.3.1.7.1
Multiply by by adding the exponents.
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Step 6.3.1.7.1.1
Move .
Step 6.3.1.7.1.2
Multiply by .
Step 6.3.1.7.2
Multiply by by adding the exponents.
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Step 6.3.1.7.2.1
Move .
Step 6.3.1.7.2.2
Multiply by .
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Step 6.3.1.7.2.2.1
Raise to the power of .
Step 6.3.1.7.2.2.2
Use the power rule to combine exponents.
Step 6.3.1.7.2.3
Add and .
Step 6.3.1.8
Apply the distributive property.
Step 6.3.1.9
Simplify.
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Step 6.3.1.9.1
Multiply by .
Step 6.3.1.9.2
Multiply by .
Step 6.3.1.9.3
Multiply by .
Step 6.3.1.10
Remove parentheses.
Step 6.3.1.11
Rewrite using the commutative property of multiplication.
Step 6.3.1.12
Multiply by .
Step 6.3.1.13
Multiply by .
Step 6.3.1.14
Multiply by by adding the exponents.
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Step 6.3.1.14.1
Move .
Step 6.3.1.14.2
Multiply by .
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Step 6.3.1.14.2.1
Raise to the power of .
Step 6.3.1.14.2.2
Use the power rule to combine exponents.
Step 6.3.1.14.3
Add and .
Step 6.3.1.15
Multiply by .
Step 6.3.1.16
Multiply by .
Step 6.3.2
Combine the opposite terms in .
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Step 6.3.2.1
Subtract from .
Step 6.3.2.2
Add and .
Step 6.3.3
Subtract from .
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Step 6.3.3.1
Move .
Step 6.3.3.2
Subtract from .
Step 6.4
Reorder terms.
Step 6.5
Factor out of .
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Step 6.5.1
Factor out of .
Step 6.5.2
Factor out of .
Step 6.5.3
Factor out of .
Step 6.6
Cancel the common factor of and .
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Step 6.6.1
Factor out of .
Step 6.6.2
Cancel the common factors.
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Step 6.6.2.1
Factor out of .
Step 6.6.2.2
Cancel the common factor.
Step 6.6.2.3
Rewrite the expression.