Calculus Examples

Find the Derivative Using Chain Rule - d/dt t(2t+1)^2
Step 1
This derivative could not be completed using the chain rule. Mathway will use another method.
Step 2
Rewrite as .
Step 3
Expand using the FOIL Method.
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Step 3.1
Apply the distributive property.
Step 3.2
Apply the distributive property.
Step 3.3
Apply the distributive property.
Step 4
Simplify and combine like terms.
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Step 4.1
Simplify each term.
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Step 4.1.1
Rewrite using the commutative property of multiplication.
Step 4.1.2
Multiply by by adding the exponents.
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Step 4.1.2.1
Move .
Step 4.1.2.2
Multiply by .
Step 4.1.3
Multiply by .
Step 4.1.4
Multiply by .
Step 4.1.5
Multiply by .
Step 4.1.6
Multiply by .
Step 4.2
Add and .
Step 5
Differentiate using the Product Rule which states that is where and .
Step 6
Differentiate.
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Step 6.1
By the Sum Rule, the derivative of with respect to is .
Step 6.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.3
Differentiate using the Power Rule which states that is where .
Step 6.4
Multiply by .
Step 6.5
Since is constant with respect to , the derivative of with respect to is .
Step 6.6
Differentiate using the Power Rule which states that is where .
Step 6.7
Multiply by .
Step 6.8
Since is constant with respect to , the derivative of with respect to is .
Step 6.9
Add and .
Step 6.10
Differentiate using the Power Rule which states that is where .
Step 6.11
Multiply by .
Step 7
Simplify.
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Step 7.1
Apply the distributive property.
Step 7.2
Combine terms.
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Step 7.2.1
Raise to the power of .
Step 7.2.2
Raise to the power of .
Step 7.2.3
Use the power rule to combine exponents.
Step 7.2.4
Add and .
Step 7.2.5
Move to the left of .
Step 7.2.6
Add and .
Step 7.2.7
Add and .