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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
Differentiate using the chain rule, which states that is where and .
Step 3.1.1
To apply the Chain Rule, set as .
Step 3.1.2
The derivative of with respect to is .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Multiply by .
Step 3.4
To multiply absolute values, multiply the terms inside each absolute value.
Step 3.5
Raise to the power of .
Step 3.6
Raise to the power of .
Step 3.7
Use the power rule to combine exponents.
Step 3.8
Add and .
Step 3.9
By the Sum Rule, the derivative of with respect to is .
Step 3.10
Since is constant with respect to , the derivative of with respect to is .
Step 3.11
Differentiate using the Power Rule which states that is where .
Step 3.12
Multiply by .
Step 3.13
Since is constant with respect to , the derivative of with respect to is .
Step 3.14
Differentiate using the Power Rule which states that is where .
Step 3.15
Multiply by .
Step 3.16
Simplify.
Step 3.16.1
Reorder the factors of .
Step 3.16.2
Factor out of .
Step 3.16.2.1
Factor out of .
Step 3.16.2.2
Factor out of .
Step 3.16.2.3
Factor out of .
Step 3.16.3
Simplify the denominator.
Step 3.16.3.1
Factor out of .
Step 3.16.3.1.1
Factor out of .
Step 3.16.3.1.2
Factor out of .
Step 3.16.3.1.3
Factor out of .
Step 3.16.3.2
Apply the product rule to .
Step 3.16.3.3
Rewrite as .
Step 3.16.3.4
Expand using the FOIL Method.
Step 3.16.3.4.1
Apply the distributive property.
Step 3.16.3.4.2
Apply the distributive property.
Step 3.16.3.4.3
Apply the distributive property.
Step 3.16.3.5
Simplify and combine like terms.
Step 3.16.3.5.1
Simplify each term.
Step 3.16.3.5.1.1
Rewrite using the commutative property of multiplication.
Step 3.16.3.5.1.2
Multiply by by adding the exponents.
Step 3.16.3.5.1.2.1
Move .
Step 3.16.3.5.1.2.2
Multiply by .
Step 3.16.3.5.1.3
Multiply by .
Step 3.16.3.5.1.4
Multiply by .
Step 3.16.3.5.1.5
Multiply by .
Step 3.16.3.5.1.6
Multiply by .
Step 3.16.3.5.2
Subtract from .
Step 3.16.3.6
Apply the distributive property.
Step 3.16.3.7
Simplify.
Step 3.16.3.7.1
Rewrite using the commutative property of multiplication.
Step 3.16.3.7.2
Rewrite using the commutative property of multiplication.
Step 3.16.3.7.3
Move to the left of .
Step 3.16.3.8
Simplify each term.
Step 3.16.3.8.1
Multiply by by adding the exponents.
Step 3.16.3.8.1.1
Move .
Step 3.16.3.8.1.2
Use the power rule to combine exponents.
Step 3.16.3.8.1.3
Add and .
Step 3.16.3.8.2
Multiply by by adding the exponents.
Step 3.16.3.8.2.1
Move .
Step 3.16.3.8.2.2
Multiply by .
Step 3.16.3.8.2.2.1
Raise to the power of .
Step 3.16.3.8.2.2.2
Use the power rule to combine exponents.
Step 3.16.3.8.2.3
Add and .
Step 3.16.3.9
Factor out of .
Step 3.16.3.9.1
Factor out of .
Step 3.16.3.9.2
Factor out of .
Step 3.16.3.9.3
Factor out of .
Step 3.16.3.9.4
Factor out of .
Step 3.16.3.9.5
Factor out of .
Step 3.16.3.10
Factor using the perfect square rule.
Step 3.16.3.10.1
Rewrite as .
Step 3.16.3.10.2
Rewrite as .
Step 3.16.3.10.3
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 3.16.3.10.4
Rewrite the polynomial.
Step 3.16.3.10.5
Factor using the perfect square trinomial rule , where and .
Step 3.16.4
Remove non-negative terms from the absolute value.
Step 3.16.5
Cancel the common factors.
Step 3.16.5.1
Factor out of .
Step 3.16.5.2
Cancel the common factor.
Step 3.16.5.3
Rewrite the expression.
Step 3.16.6
Cancel the common factor of and .
Step 3.16.6.1
Multiply by .
Step 3.16.6.2
Cancel the common factors.
Step 3.16.6.2.1
Factor out of .
Step 3.16.6.2.2
Cancel the common factor.
Step 3.16.6.2.3
Rewrite the expression.
Step 3.16.7
Multiply by .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .