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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Step 3.1
Rewrite as .
Step 3.2
Expand using the FOIL Method.
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.
Step 3.3.1
Simplify each term.
Step 3.3.1.1
Rewrite using the commutative property of multiplication.
Step 3.3.1.2
Multiply by by adding the exponents.
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 .
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.
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.
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.
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 .
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.
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.
Step 3.7.8.1
Simplify each term.
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.
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.
Step 3.7.9.1
Expand using the FOIL Method.
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.
Step 3.7.9.2.1
Simplify each term.
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.
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.
Step 3.7.14.1
Rewrite using the commutative property of multiplication.
Step 3.7.14.2
Multiply by by adding the exponents.
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.
Step 3.7.14.5.1
Move .
Step 3.7.14.5.2
Multiply by .
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.
Step 3.7.14.9.1
Move .
Step 3.7.14.9.2
Multiply by .
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.
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 .