Calculus Examples

Find the Derivative - d/dx y=(x^2-4x-6)(x^3-5x^2-3x)
Step 1
Differentiate using the Product 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
Differentiate using the Power Rule which states that is where .
Step 2.8
Multiply by .
Step 2.9
By the Sum Rule, the derivative of with respect to is .
Step 2.10
Differentiate using the Power Rule which states that is where .
Step 2.11
Since is constant with respect to , the derivative of with respect to is .
Step 2.12
Differentiate using the Power Rule which states that is where .
Step 2.13
Multiply by .
Step 2.14
Since is constant with respect to , the derivative of with respect to is .
Step 2.15
Add and .
Step 3
Simplify.
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Step 3.1
Factor out of .
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Step 3.1.1
Factor out of .
Step 3.1.2
Factor out of .
Step 3.1.3
Factor out of .
Step 3.2
Multiply by .
Step 3.3
Reorder terms.
Step 3.4
Simplify each term.
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Step 3.4.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.4.2
Simplify each term.
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Step 3.4.2.1
Multiply by by adding the exponents.
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Step 3.4.2.1.1
Move .
Step 3.4.2.1.2
Use the power rule to combine exponents.
Step 3.4.2.1.3
Add and .
Step 3.4.2.2
Rewrite using the commutative property of multiplication.
Step 3.4.2.3
Multiply by by adding the exponents.
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Step 3.4.2.3.1
Move .
Step 3.4.2.3.2
Multiply by .
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Step 3.4.2.3.2.1
Raise to the power of .
Step 3.4.2.3.2.2
Use the power rule to combine exponents.
Step 3.4.2.3.3
Add and .
Step 3.4.2.4
Multiply by .
Step 3.4.2.5
Multiply by .
Step 3.4.2.6
Multiply by by adding the exponents.
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Step 3.4.2.6.1
Move .
Step 3.4.2.6.2
Multiply by .
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Step 3.4.2.6.2.1
Raise to the power of .
Step 3.4.2.6.2.2
Use the power rule to combine exponents.
Step 3.4.2.6.3
Add and .
Step 3.4.2.7
Rewrite using the commutative property of multiplication.
Step 3.4.2.8
Multiply by by adding the exponents.
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Step 3.4.2.8.1
Move .
Step 3.4.2.8.2
Multiply by .
Step 3.4.2.9
Multiply by .
Step 3.4.2.10
Multiply by .
Step 3.4.2.11
Multiply by .
Step 3.4.2.12
Multiply by .
Step 3.4.3
Subtract from .
Step 3.4.4
Add and .
Step 3.4.5
Subtract from .
Step 3.4.6
Add and .
Step 3.4.7
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.4.8
Simplify each term.
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Step 3.4.8.1
Multiply by by adding the exponents.
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Step 3.4.8.1.1
Move .
Step 3.4.8.1.2
Multiply by .
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Step 3.4.8.1.2.1
Raise to the power of .
Step 3.4.8.1.2.2
Use the power rule to combine exponents.
Step 3.4.8.1.3
Add and .
Step 3.4.8.2
Rewrite using the commutative property of multiplication.
Step 3.4.8.3
Multiply by by adding the exponents.
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Step 3.4.8.3.1
Move .
Step 3.4.8.3.2
Multiply by .
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Step 3.4.8.3.2.1
Raise to the power of .
Step 3.4.8.3.2.2
Use the power rule to combine exponents.
Step 3.4.8.3.3
Add and .
Step 3.4.8.4
Multiply by .
Step 3.4.8.5
Rewrite using the commutative property of multiplication.
Step 3.4.8.6
Multiply by by adding the exponents.
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Step 3.4.8.6.1
Move .
Step 3.4.8.6.2
Multiply by .
Step 3.4.8.7
Multiply by .
Step 3.4.8.8
Multiply by .
Step 3.4.8.9
Multiply by .
Step 3.4.9
Subtract from .
Step 3.4.10
Add and .
Step 3.5
Add and .
Step 3.6
Subtract from .
Step 3.7
Add and .
Step 3.8
Add and .