Calculus Examples

Find dy/dx y=(3x+4)^2(2x-5)^3
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Rewrite as .
Step 3.2
Expand using the FOIL Method.
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Step 3.2.1
Apply the distributive property.
Step 3.2.2
Apply the distributive property.
Step 3.2.3
Apply the distributive property.
Step 3.3
Simplify and combine like terms.
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Step 3.3.1
Simplify each term.
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Step 3.3.1.1
Rewrite using the commutative property of multiplication.
Step 3.3.1.2
Multiply by by adding the exponents.
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Step 3.3.1.2.1
Move .
Step 3.3.1.2.2
Multiply by .
Step 3.3.1.3
Multiply by .
Step 3.3.1.4
Multiply by .
Step 3.3.1.5
Multiply by .
Step 3.3.1.6
Multiply by .
Step 3.3.2
Add and .
Step 3.4
Differentiate using the Product Rule which states that is where and .
Step 3.5
Differentiate using the chain rule, which states that is where and .
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Step 3.5.1
To apply the Chain Rule, set as .
Step 3.5.2
Differentiate using the Power Rule which states that is where .
Step 3.5.3
Replace all occurrences of with .
Step 3.6
Differentiate.
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Step 3.6.1
Move to the left of .
Step 3.6.2
By the Sum Rule, the derivative of with respect to is .
Step 3.6.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.4
Differentiate using the Power Rule which states that is where .
Step 3.6.5
Multiply by .
Step 3.6.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.7
Simplify the expression.
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Step 3.6.7.1
Add and .
Step 3.6.7.2
Multiply by .
Step 3.6.8
By the Sum Rule, the derivative of with respect to is .
Step 3.6.9
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.10
Differentiate using the Power Rule which states that is where .
Step 3.6.11
Multiply by .
Step 3.6.12
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.13
Differentiate using the Power Rule which states that is where .
Step 3.6.14
Multiply by .
Step 3.6.15
Since is constant with respect to , the derivative of with respect to is .
Step 3.6.16
Add and .
Step 3.7
Simplify.
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Step 3.7.1
Apply the distributive property.
Step 3.7.2
Multiply by .
Step 3.7.3
Multiply by .
Step 3.7.4
Multiply by .
Step 3.7.5
Factor out of .
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Step 3.7.5.1
Factor out of .
Step 3.7.5.2
Factor out of .
Step 3.7.5.3
Factor out of .
Step 3.7.6
Rewrite as .
Step 3.7.7
Expand using the FOIL Method.
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Step 3.7.7.1
Apply the distributive property.
Step 3.7.7.2
Apply the distributive property.
Step 3.7.7.3
Apply the distributive property.
Step 3.7.8
Simplify and combine like terms.
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Step 3.7.8.1
Simplify each term.
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Step 3.7.8.1.1
Rewrite using the commutative property of multiplication.
Step 3.7.8.1.2
Multiply by by adding the exponents.
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Step 3.7.8.1.2.1
Move .
Step 3.7.8.1.2.2
Multiply by .
Step 3.7.8.1.3
Multiply by .
Step 3.7.8.1.4
Multiply by .
Step 3.7.8.1.5
Multiply by .
Step 3.7.8.1.6
Multiply by .
Step 3.7.8.2
Subtract from .
Step 3.7.9
Simplify each term.
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Step 3.7.9.1
Expand using the FOIL Method.
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Step 3.7.9.1.1
Apply the distributive property.
Step 3.7.9.1.2
Apply the distributive property.
Step 3.7.9.1.3
Apply the distributive property.
Step 3.7.9.2
Simplify and combine like terms.
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Step 3.7.9.2.1
Simplify each term.
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Step 3.7.9.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.7.9.2.1.2
Multiply by by adding the exponents.
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Step 3.7.9.2.1.2.1
Move .
Step 3.7.9.2.1.2.2
Multiply by .
Step 3.7.9.2.1.3
Multiply by .
Step 3.7.9.2.1.4
Multiply by .
Step 3.7.9.2.1.5
Multiply by .
Step 3.7.9.2.1.6
Multiply by .
Step 3.7.9.2.2
Subtract from .
Step 3.7.10
Add and .
Step 3.7.11
Subtract from .
Step 3.7.12
Subtract from .
Step 3.7.13
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.7.14
Simplify each term.
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Step 3.7.14.1
Rewrite using the commutative property of multiplication.
Step 3.7.14.2
Multiply by by adding the exponents.
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Step 3.7.14.2.1
Move .
Step 3.7.14.2.2
Use the power rule to combine exponents.
Step 3.7.14.2.3
Add and .
Step 3.7.14.3
Multiply by .
Step 3.7.14.4
Rewrite using the commutative property of multiplication.
Step 3.7.14.5
Multiply by by adding the exponents.
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Step 3.7.14.5.1
Move .
Step 3.7.14.5.2
Multiply by .
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Step 3.7.14.5.2.1
Raise to the power of .
Step 3.7.14.5.2.2
Use the power rule to combine exponents.
Step 3.7.14.5.3
Add and .
Step 3.7.14.6
Multiply by .
Step 3.7.14.7
Multiply by .
Step 3.7.14.8
Rewrite using the commutative property of multiplication.
Step 3.7.14.9
Multiply by by adding the exponents.
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Step 3.7.14.9.1
Move .
Step 3.7.14.9.2
Multiply by .
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Step 3.7.14.9.2.1
Raise to the power of .
Step 3.7.14.9.2.2
Use the power rule to combine exponents.
Step 3.7.14.9.3
Add and .
Step 3.7.14.10
Multiply by .
Step 3.7.14.11
Rewrite using the commutative property of multiplication.
Step 3.7.14.12
Multiply by by adding the exponents.
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Step 3.7.14.12.1
Move .
Step 3.7.14.12.2
Multiply by .
Step 3.7.14.13
Multiply by .
Step 3.7.14.14
Multiply by .
Step 3.7.14.15
Multiply by .
Step 3.7.14.16
Multiply by .
Step 3.7.14.17
Multiply by .
Step 3.7.15
Subtract from .
Step 3.7.16
Subtract from .
Step 3.7.17
Add and .
Step 3.7.18
Add and .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .