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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
Differentiate using the Power Rule which states that is where .
Step 3.1.3
Replace all occurrences of with .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
Differentiate.
Step 3.3.1
Differentiate using the Power Rule which states that is where .
Step 3.3.2
Move to the left of .
Step 3.3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.3.4
Differentiate using the Power Rule which states that is where .
Step 3.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.6
Differentiate using the Power Rule which states that is where .
Step 3.3.7
Combine fractions.
Step 3.3.7.1
Multiply by .
Step 3.3.7.2
Combine and .
Step 3.3.7.3
Move to the left of .
Step 3.4
Simplify.
Step 3.4.1
Apply the product rule to .
Step 3.4.2
Apply the distributive property.
Step 3.4.3
Apply the distributive property.
Step 3.4.4
Apply the distributive property.
Step 3.4.5
Apply the distributive property.
Step 3.4.6
Combine terms.
Step 3.4.6.1
Raise to the power of .
Step 3.4.6.2
Use the power rule to combine exponents.
Step 3.4.6.3
Add and .
Step 3.4.6.4
Multiply by .
Step 3.4.6.5
Multiply by .
Step 3.4.6.6
Raise to the power of .
Step 3.4.6.7
Raise to the power of .
Step 3.4.6.8
Use the power rule to combine exponents.
Step 3.4.6.9
Add and .
Step 3.4.6.10
Multiply by .
Step 3.4.6.11
Multiply by .
Step 3.4.6.12
Multiply by by adding the exponents.
Step 3.4.6.12.1
Move .
Step 3.4.6.12.2
Use the power rule to combine exponents.
Step 3.4.6.12.3
Add and .
Step 3.4.6.13
Multiply by .
Step 3.4.6.14
Multiply by .
Step 3.4.6.15
Multiply by .
Step 3.4.6.16
Subtract from .
Step 3.4.6.17
Add and .
Step 3.4.6.18
Multiply the exponents in .
Step 3.4.6.18.1
Apply the power rule and multiply exponents, .
Step 3.4.6.18.2
Multiply by .
Step 3.4.6.19
Multiply by .
Step 3.4.6.20
Multiply by by adding the exponents.
Step 3.4.6.20.1
Use the power rule to combine exponents.
Step 3.4.6.20.2
Add and .
Step 3.4.7
Reorder terms.
Step 3.4.8
Simplify the numerator.
Step 3.4.8.1
Factor out of .
Step 3.4.8.1.1
Factor out of .
Step 3.4.8.1.2
Factor out of .
Step 3.4.8.1.3
Factor out of .
Step 3.4.8.2
Multiply by by adding the exponents.
Step 3.4.8.2.1
Move .
Step 3.4.8.2.2
Use the power rule to combine exponents.
Step 3.4.8.2.3
Add and .
Step 3.4.9
Simplify the denominator.
Step 3.4.9.1
Factor out of .
Step 3.4.9.1.1
Factor out of .
Step 3.4.9.1.2
Factor out of .
Step 3.4.9.1.3
Factor out of .
Step 3.4.9.2
Rewrite as .
Step 3.4.9.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.4.9.4
Apply the product rule to .
Step 3.4.9.5
Apply the distributive property.
Step 3.4.9.6
Multiply by .
Step 3.4.9.7
Move to the left of .
Step 3.4.9.8
Use the Binomial Theorem.
Step 3.4.9.9
Simplify each term.
Step 3.4.9.9.1
Multiply the exponents in .
Step 3.4.9.9.1.1
Apply the power rule and multiply exponents, .
Step 3.4.9.9.1.2
Multiply by .
Step 3.4.9.9.2
Rewrite using the commutative property of multiplication.
Step 3.4.9.9.3
Multiply by .
Step 3.4.9.9.4
Multiply the exponents in .
Step 3.4.9.9.4.1
Apply the power rule and multiply exponents, .
Step 3.4.9.9.4.2
Multiply by .
Step 3.4.9.9.5
Multiply by by adding the exponents.
Step 3.4.9.9.5.1
Move .
Step 3.4.9.9.5.2
Multiply by .
Step 3.4.9.9.5.2.1
Raise to the power of .
Step 3.4.9.9.5.2.2
Use the power rule to combine exponents.
Step 3.4.9.9.5.3
Add and .
Step 3.4.9.9.6
Multiply the exponents in .
Step 3.4.9.9.6.1
Apply the power rule and multiply exponents, .
Step 3.4.9.9.6.2
Multiply by .
Step 3.4.9.9.7
Apply the product rule to .
Step 3.4.9.9.8
Rewrite using the commutative property of multiplication.
Step 3.4.9.9.9
Multiply by by adding the exponents.
Step 3.4.9.9.9.1
Move .
Step 3.4.9.9.9.2
Use the power rule to combine exponents.
Step 3.4.9.9.9.3
Add and .
Step 3.4.9.9.10
Raise to the power of .
Step 3.4.9.9.11
Multiply by .
Step 3.4.9.9.12
Apply the product rule to .
Step 3.4.9.9.13
Rewrite using the commutative property of multiplication.
Step 3.4.9.9.14
Multiply by by adding the exponents.
Step 3.4.9.9.14.1
Move .
Step 3.4.9.9.14.2
Use the power rule to combine exponents.
Step 3.4.9.9.14.3
Add and .
Step 3.4.9.9.15
Raise to the power of .
Step 3.4.9.9.16
Multiply by .
Step 3.4.9.9.17
Apply the product rule to .
Step 3.4.9.9.18
Raise to the power of .
Step 3.4.9.10
Factor out of .
Step 3.4.9.10.1
Factor out of .
Step 3.4.9.10.2
Factor out of .
Step 3.4.9.10.3
Factor out of .
Step 3.4.9.10.4
Factor out of .
Step 3.4.9.10.5
Factor out of .
Step 3.4.9.10.6
Factor out of .
Step 3.4.9.10.7
Factor out of .
Step 3.4.9.10.8
Factor out of .
Step 3.4.9.10.9
Factor out of .
Step 3.4.9.11
Make each term match the terms from the binomial theorem formula.
Step 3.4.9.12
Factor using the binomial theorem.
Step 3.4.10
Cancel the common factor of and .
Step 3.4.10.1
Factor out of .
Step 3.4.10.2
Cancel the common factors.
Step 3.4.10.2.1
Factor out of .
Step 3.4.10.2.2
Cancel the common factor.
Step 3.4.10.2.3
Rewrite the expression.
Step 3.4.11
Move to the left of .
Step 3.4.12
Factor out of .
Step 3.4.13
Rewrite as .
Step 3.4.14
Factor out of .
Step 3.4.15
Rewrite as .
Step 3.4.16
Move the negative in front of the fraction.
Step 3.4.17
Reorder factors in .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .