Calculus Examples

Find the Derivative - d/dx y=2/3*(x^3(4x^5-13)^3)
Step 1
Differentiate using the Constant Multiple Rule.
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Step 1.1
Combine and .
Step 1.2
Combine fractions.
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Step 1.2.1
Combine and .
Step 1.2.2
Move to the left of .
Step 1.3
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Product Rule which states that is where and .
Step 3
Differentiate using the chain rule, which states that is where and .
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Step 3.1
To apply the Chain Rule, set as .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Replace all occurrences of with .
Step 4
Differentiate.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.3
Differentiate using the Power Rule which states that is where .
Step 4.4
Multiply by .
Step 4.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.6
Simplify the expression.
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Step 4.6.1
Add and .
Step 4.6.2
Multiply by .
Step 5
Multiply by by adding the exponents.
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Step 5.1
Move .
Step 5.2
Use the power rule to combine exponents.
Step 5.3
Add and .
Step 6
Differentiate using the Power Rule which states that is where .
Step 7
Move to the left of .
Step 8
Simplify.
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Step 8.1
Apply the distributive property.
Step 8.2
Combine and .
Step 8.3
Combine and .
Step 8.4
Multiply by .
Step 8.5
Combine and .
Step 8.6
Cancel the common factor of and .
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Step 8.6.1
Factor out of .
Step 8.6.2
Cancel the common factors.
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Step 8.6.2.1
Factor out of .
Step 8.6.2.2
Cancel the common factor.
Step 8.6.2.3
Rewrite the expression.
Step 8.6.2.4
Divide by .
Step 8.7
Combine and .
Step 8.8
Multiply by .
Step 8.9
Combine and .
Step 8.10
Combine and .
Step 8.11
Move to the left of .
Step 8.12
Cancel the common factor of and .
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Step 8.12.1
Factor out of .
Step 8.12.2
Cancel the common factors.
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Step 8.12.2.1
Factor out of .
Step 8.12.2.2
Cancel the common factor.
Step 8.12.2.3
Rewrite the expression.
Step 8.12.2.4
Divide by .
Step 8.13
Factor out of .
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Step 8.13.1
Factor out of .
Step 8.13.2
Factor out of .
Step 8.13.3
Factor out of .
Step 8.14
Add and .
Step 8.15
Rewrite as .
Step 8.16
Expand using the FOIL Method.
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Step 8.16.1
Apply the distributive property.
Step 8.16.2
Apply the distributive property.
Step 8.16.3
Apply the distributive property.
Step 8.17
Simplify and combine like terms.
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Step 8.17.1
Simplify each term.
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Step 8.17.1.1
Rewrite using the commutative property of multiplication.
Step 8.17.1.2
Multiply by by adding the exponents.
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Step 8.17.1.2.1
Move .
Step 8.17.1.2.2
Use the power rule to combine exponents.
Step 8.17.1.2.3
Add and .
Step 8.17.1.3
Multiply by .
Step 8.17.1.4
Multiply by .
Step 8.17.1.5
Multiply by .
Step 8.17.1.6
Multiply by .
Step 8.17.2
Subtract from .
Step 8.18
Apply the distributive property.
Step 8.19
Simplify.
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Step 8.19.1
Rewrite using the commutative property of multiplication.
Step 8.19.2
Rewrite using the commutative property of multiplication.
Step 8.19.3
Multiply by .
Step 8.20
Simplify each term.
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Step 8.20.1
Multiply by by adding the exponents.
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Step 8.20.1.1
Move .
Step 8.20.1.2
Use the power rule to combine exponents.
Step 8.20.1.3
Add and .
Step 8.20.2
Multiply by .
Step 8.20.3
Multiply by by adding the exponents.
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Step 8.20.3.1
Move .
Step 8.20.3.2
Use the power rule to combine exponents.
Step 8.20.3.3
Add and .
Step 8.20.4
Multiply by .
Step 8.21
Expand by multiplying each term in the first expression by each term in the second expression.
Step 8.22
Simplify each term.
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Step 8.22.1
Rewrite using the commutative property of multiplication.
Step 8.22.2
Multiply by by adding the exponents.
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Step 8.22.2.1
Move .
Step 8.22.2.2
Use the power rule to combine exponents.
Step 8.22.2.3
Add and .
Step 8.22.3
Multiply by .
Step 8.22.4
Multiply by .
Step 8.22.5
Rewrite using the commutative property of multiplication.
Step 8.22.6
Multiply by by adding the exponents.
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Step 8.22.6.1
Move .
Step 8.22.6.2
Use the power rule to combine exponents.
Step 8.22.6.3
Add and .
Step 8.22.7
Multiply by .
Step 8.22.8
Multiply by .
Step 8.22.9
Rewrite using the commutative property of multiplication.
Step 8.22.10
Multiply by by adding the exponents.
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Step 8.22.10.1
Move .
Step 8.22.10.2
Use the power rule to combine exponents.
Step 8.22.10.3
Add and .
Step 8.22.11
Multiply by .
Step 8.22.12
Multiply by .
Step 8.23
Subtract from .
Step 8.24
Add and .