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Calculus Examples
Step 1
By the Sum Rule, the derivative of with respect to is .
Step 2
Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
The derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Combine and .
Step 2.5
Multiply by .
Step 3
Step 3.1
Use to rewrite as .
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
Differentiate using the Power Rule which states that is where .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
By the Sum Rule, the derivative of with respect to is .
Step 3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
To write as a fraction with a common denominator, multiply by .
Step 3.8
Combine and .
Step 3.9
Combine the numerators over the common denominator.
Step 3.10
Simplify the numerator.
Step 3.10.1
Multiply by .
Step 3.10.2
Subtract from .
Step 3.11
Move the negative in front of the fraction.
Step 3.12
Multiply by .
Step 3.13
Subtract from .
Step 3.14
Combine and .
Step 3.15
Combine and .
Step 3.16
Combine and .
Step 3.17
Move to the denominator using the negative exponent rule .
Step 3.18
Factor out of .
Step 3.19
Cancel the common factors.
Step 3.19.1
Factor out of .
Step 3.19.2
Cancel the common factor.
Step 3.19.3
Rewrite the expression.
Step 3.20
Move the negative in front of the fraction.
Step 4
Step 4.1
Reorder terms.
Step 4.2
Simplify each term.
Step 4.2.1
Simplify the denominator.
Step 4.2.1.1
Rewrite as .
Step 4.2.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.2
Multiply by .
Step 4.2.3
Combine and simplify the denominator.
Step 4.2.3.1
Multiply by .
Step 4.2.3.2
Raise to the power of .
Step 4.2.3.3
Raise to the power of .
Step 4.2.3.4
Use the power rule to combine exponents.
Step 4.2.3.5
Add and .
Step 4.2.3.6
Rewrite as .
Step 4.2.3.6.1
Use to rewrite as .
Step 4.2.3.6.2
Apply the power rule and multiply exponents, .
Step 4.2.3.6.3
Combine and .
Step 4.2.3.6.4
Cancel the common factor of .
Step 4.2.3.6.4.1
Cancel the common factor.
Step 4.2.3.6.4.2
Rewrite the expression.
Step 4.2.3.6.5
Simplify.
Step 4.3
To write as a fraction with a common denominator, multiply by .
Step 4.4
To write as a fraction with a common denominator, multiply by .
Step 4.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.5.1
Multiply by .
Step 4.5.2
Multiply by .
Step 4.5.3
Reorder the factors of .
Step 4.5.4
Reorder the factors of .
Step 4.6
Combine the numerators over the common denominator.
Step 4.7
Simplify the numerator.
Step 4.7.1
Factor out of .
Step 4.7.1.1
Factor out of .
Step 4.7.1.2
Factor out of .
Step 4.7.1.3
Factor out of .
Step 4.7.2
Expand using the FOIL Method.
Step 4.7.2.1
Apply the distributive property.
Step 4.7.2.2
Apply the distributive property.
Step 4.7.2.3
Apply the distributive property.
Step 4.7.3
Simplify and combine like terms.
Step 4.7.3.1
Simplify each term.
Step 4.7.3.1.1
Multiply by .
Step 4.7.3.1.2
Multiply by .
Step 4.7.3.1.3
Multiply by .
Step 4.7.3.1.4
Rewrite using the commutative property of multiplication.
Step 4.7.3.1.5
Multiply by by adding the exponents.
Step 4.7.3.1.5.1
Move .
Step 4.7.3.1.5.2
Multiply by .
Step 4.7.3.2
Add and .
Step 4.7.3.3
Add and .
Step 4.7.4
Apply the distributive property.
Step 4.7.5
Multiply by .
Step 4.7.6
Multiply .
Step 4.7.6.1
Multiply by .
Step 4.7.6.2
Multiply by .
Step 4.8
To write as a fraction with a common denominator, multiply by .
Step 4.9
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 4.9.1
Combine and .
Step 4.9.2
Reorder the factors of .
Step 4.10
Combine the numerators over the common denominator.
Step 4.11
Simplify the numerator.
Step 4.11.1
Use to rewrite as .
Step 4.11.2
Simplify each term.
Step 4.11.2.1
Expand using the FOIL Method.
Step 4.11.2.1.1
Apply the distributive property.
Step 4.11.2.1.2
Apply the distributive property.
Step 4.11.2.1.3
Apply the distributive property.
Step 4.11.2.2
Simplify and combine like terms.
Step 4.11.2.2.1
Simplify each term.
Step 4.11.2.2.1.1
Multiply by .
Step 4.11.2.2.1.2
Multiply by .
Step 4.11.2.2.1.3
Multiply by .
Step 4.11.2.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.11.2.2.1.5
Multiply by by adding the exponents.
Step 4.11.2.2.1.5.1
Move .
Step 4.11.2.2.1.5.2
Multiply by .
Step 4.11.2.2.2
Add and .
Step 4.11.2.2.3
Add and .
Step 4.11.2.3
Multiply by by adding the exponents.
Step 4.11.2.3.1
Use the power rule to combine exponents.
Step 4.11.2.3.2
Combine the numerators over the common denominator.
Step 4.11.2.3.3
Add and .
Step 4.11.2.3.4
Divide by .
Step 4.11.2.4
Simplify .
Step 4.11.3
Combine the opposite terms in .
Step 4.11.3.1
Subtract from .
Step 4.11.3.2
Add and .
Step 4.11.3.3
Add and .
Step 4.11.4
Multiply by .
Step 4.11.5
Expand using the FOIL Method.
Step 4.11.5.1
Apply the distributive property.
Step 4.11.5.2
Apply the distributive property.
Step 4.11.5.3
Apply the distributive property.
Step 4.11.6
Simplify and combine like terms.
Step 4.11.6.1
Simplify each term.
Step 4.11.6.1.1
Multiply by .
Step 4.11.6.1.2
Multiply by .
Step 4.11.6.1.3
Multiply by .
Step 4.11.6.1.4
Rewrite using the commutative property of multiplication.
Step 4.11.6.1.5
Multiply by by adding the exponents.
Step 4.11.6.1.5.1
Move .
Step 4.11.6.1.5.2
Multiply by .
Step 4.11.6.2
Add and .
Step 4.11.6.3
Add and .
Step 4.11.7
Multiply by by adding the exponents.
Step 4.11.7.1
Multiply by .
Step 4.11.7.1.1
Raise to the power of .
Step 4.11.7.1.2
Use the power rule to combine exponents.
Step 4.11.7.2
Write as a fraction with a common denominator.
Step 4.11.7.3
Combine the numerators over the common denominator.
Step 4.11.7.4
Add and .
Step 4.11.8
Add and .
Step 4.12
Move to the numerator using the negative exponent rule .
Step 4.13
Simplify the numerator.
Step 4.13.1
Multiply by by adding the exponents.
Step 4.13.1.1
Move .
Step 4.13.1.2
Use the power rule to combine exponents.
Step 4.13.1.3
Combine the numerators over the common denominator.
Step 4.13.1.4
Add and .
Step 4.13.1.5
Divide by .
Step 4.13.2
Simplify .
Step 4.14
Simplify the numerator.
Step 4.14.1
Rewrite as .
Step 4.14.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.15
Cancel the common factor of .
Step 4.15.1
Cancel the common factor.
Step 4.15.2
Rewrite the expression.
Step 4.16
Cancel the common factor of .
Step 4.16.1
Cancel the common factor.
Step 4.16.2
Divide by .