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Calculus Examples
Step 1
Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
Differentiate.
Step 1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.2.2
Multiply by .
Step 1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.6
Simplify the expression.
Step 1.2.6.1
Add and .
Step 1.2.6.2
Multiply by .
Step 1.3
Raise to the power of .
Step 1.4
Raise to the power of .
Step 1.5
Use the power rule to combine exponents.
Step 1.6
Add and .
Step 1.7
Subtract from .
Step 2
Step 2.1
Differentiate using the Quotient Rule which states that is where and .
Step 2.2
Differentiate.
Step 2.2.1
Multiply the exponents in .
Step 2.2.1.1
Apply the power rule and multiply exponents, .
Step 2.2.1.2
Multiply by .
Step 2.2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.4
Differentiate using the Power Rule which states that is where .
Step 2.2.5
Multiply by .
Step 2.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.7
Add and .
Step 2.3
Differentiate using the chain rule, which states that is where and .
Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Differentiate.
Step 2.4.1
Multiply by .
Step 2.4.2
By the Sum Rule, the derivative of with respect to is .
Step 2.4.3
Differentiate using the Power Rule which states that is where .
Step 2.4.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.5
Simplify the expression.
Step 2.4.5.1
Add and .
Step 2.4.5.2
Move to the left of .
Step 2.4.5.3
Multiply by .
Step 2.5
Simplify.
Step 2.5.1
Apply the distributive property.
Step 2.5.2
Apply the distributive property.
Step 2.5.3
Simplify the numerator.
Step 2.5.3.1
Simplify each term.
Step 2.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 2.5.3.1.2
Rewrite as .
Step 2.5.3.1.3
Expand using the FOIL Method.
Step 2.5.3.1.3.1
Apply the distributive property.
Step 2.5.3.1.3.2
Apply the distributive property.
Step 2.5.3.1.3.3
Apply the distributive property.
Step 2.5.3.1.4
Simplify and combine like terms.
Step 2.5.3.1.4.1
Simplify each term.
Step 2.5.3.1.4.1.1
Multiply by by adding the exponents.
Step 2.5.3.1.4.1.1.1
Use the power rule to combine exponents.
Step 2.5.3.1.4.1.1.2
Add and .
Step 2.5.3.1.4.1.2
Multiply by .
Step 2.5.3.1.4.1.3
Multiply by .
Step 2.5.3.1.4.1.4
Multiply by .
Step 2.5.3.1.4.2
Add and .
Step 2.5.3.1.5
Apply the distributive property.
Step 2.5.3.1.6
Simplify.
Step 2.5.3.1.6.1
Multiply by .
Step 2.5.3.1.6.2
Multiply by .
Step 2.5.3.1.7
Apply the distributive property.
Step 2.5.3.1.8
Simplify.
Step 2.5.3.1.8.1
Multiply by by adding the exponents.
Step 2.5.3.1.8.1.1
Move .
Step 2.5.3.1.8.1.2
Multiply by .
Step 2.5.3.1.8.1.2.1
Raise to the power of .
Step 2.5.3.1.8.1.2.2
Use the power rule to combine exponents.
Step 2.5.3.1.8.1.3
Add and .
Step 2.5.3.1.8.2
Multiply by by adding the exponents.
Step 2.5.3.1.8.2.1
Move .
Step 2.5.3.1.8.2.2
Multiply by .
Step 2.5.3.1.8.2.2.1
Raise to the power of .
Step 2.5.3.1.8.2.2.2
Use the power rule to combine exponents.
Step 2.5.3.1.8.2.3
Add and .
Step 2.5.3.1.9
Simplify each term.
Step 2.5.3.1.9.1
Multiply by .
Step 2.5.3.1.9.2
Multiply by .
Step 2.5.3.1.10
Simplify each term.
Step 2.5.3.1.10.1
Multiply by by adding the exponents.
Step 2.5.3.1.10.1.1
Multiply by .
Step 2.5.3.1.10.1.1.1
Raise to the power of .
Step 2.5.3.1.10.1.1.2
Use the power rule to combine exponents.
Step 2.5.3.1.10.1.2
Add and .
Step 2.5.3.1.10.2
Multiply by .
Step 2.5.3.1.11
Expand using the FOIL Method.
Step 2.5.3.1.11.1
Apply the distributive property.
Step 2.5.3.1.11.2
Apply the distributive property.
Step 2.5.3.1.11.3
Apply the distributive property.
Step 2.5.3.1.12
Simplify and combine like terms.
Step 2.5.3.1.12.1
Simplify each term.
Step 2.5.3.1.12.1.1
Multiply by by adding the exponents.
Step 2.5.3.1.12.1.1.1
Move .
Step 2.5.3.1.12.1.1.2
Use the power rule to combine exponents.
Step 2.5.3.1.12.1.1.3
Add and .
Step 2.5.3.1.12.1.2
Multiply by by adding the exponents.
Step 2.5.3.1.12.1.2.1
Move .
Step 2.5.3.1.12.1.2.2
Multiply by .
Step 2.5.3.1.12.1.2.2.1
Raise to the power of .
Step 2.5.3.1.12.1.2.2.2
Use the power rule to combine exponents.
Step 2.5.3.1.12.1.2.3
Add and .
Step 2.5.3.1.12.2
Subtract from .
Step 2.5.3.1.12.3
Add and .
Step 2.5.3.2
Add and .
Step 2.5.3.3
Subtract from .
Step 2.5.4
Simplify the numerator.
Step 2.5.4.1
Factor out of .
Step 2.5.4.1.1
Factor out of .
Step 2.5.4.1.2
Factor out of .
Step 2.5.4.1.3
Factor out of .
Step 2.5.4.1.4
Factor out of .
Step 2.5.4.1.5
Factor out of .
Step 2.5.4.2
Rewrite as .
Step 2.5.4.3
Let . Substitute for all occurrences of .
Step 2.5.4.4
Factor using the AC method.
Step 2.5.4.4.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.5.4.4.2
Write the factored form using these integers.
Step 2.5.4.5
Replace all occurrences of with .
Step 2.5.5
Cancel the common factor of and .
Step 2.5.5.1
Factor out of .
Step 2.5.5.2
Cancel the common factors.
Step 2.5.5.2.1
Factor out of .
Step 2.5.5.2.2
Cancel the common factor.
Step 2.5.5.2.3
Rewrite the expression.
Step 3
Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
Multiply the exponents in .
Step 3.3.1
Apply the power rule and multiply exponents, .
Step 3.3.2
Multiply by .
Step 3.4
Differentiate using the Product Rule which states that is where and .
Step 3.5
Differentiate.
Step 3.5.1
By the Sum Rule, the derivative of with respect to is .
Step 3.5.2
Differentiate using the Power Rule which states that is where .
Step 3.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.4
Add and .
Step 3.6
Raise to the power of .
Step 3.7
Raise to the power of .
Step 3.8
Use the power rule to combine exponents.
Step 3.9
Differentiate using the Power Rule.
Step 3.9.1
Add and .
Step 3.9.2
Differentiate using the Power Rule which states that is where .
Step 3.9.3
Simplify by adding terms.
Step 3.9.3.1
Multiply by .
Step 3.9.3.2
Add and .
Step 3.10
Differentiate using the chain rule, which states that is where and .
Step 3.10.1
To apply the Chain Rule, set as .
Step 3.10.2
Differentiate using the Power Rule which states that is where .
Step 3.10.3
Replace all occurrences of with .
Step 3.11
Simplify with factoring out.
Step 3.11.1
Multiply by .
Step 3.11.2
Factor out of .
Step 3.11.2.1
Factor out of .
Step 3.11.2.2
Factor out of .
Step 3.11.2.3
Factor out of .
Step 3.12
Cancel the common factors.
Step 3.12.1
Factor out of .
Step 3.12.2
Cancel the common factor.
Step 3.12.3
Rewrite the expression.
Step 3.13
By the Sum Rule, the derivative of with respect to is .
Step 3.14
Differentiate using the Power Rule which states that is where .
Step 3.15
Since is constant with respect to , the derivative of with respect to is .
Step 3.16
Simplify the expression.
Step 3.16.1
Add and .
Step 3.16.2
Multiply by .
Step 3.17
Raise to the power of .
Step 3.18
Raise to the power of .
Step 3.19
Use the power rule to combine exponents.
Step 3.20
Add and .
Step 3.21
Combine and .
Step 3.22
Simplify.
Step 3.22.1
Apply the distributive property.
Step 3.22.2
Apply the distributive property.
Step 3.22.3
Simplify the numerator.
Step 3.22.3.1
Simplify each term.
Step 3.22.3.1.1
Expand using the FOIL Method.
Step 3.22.3.1.1.1
Apply the distributive property.
Step 3.22.3.1.1.2
Apply the distributive property.
Step 3.22.3.1.1.3
Apply the distributive property.
Step 3.22.3.1.2
Simplify and combine like terms.
Step 3.22.3.1.2.1
Simplify each term.
Step 3.22.3.1.2.1.1
Rewrite using the commutative property of multiplication.
Step 3.22.3.1.2.1.2
Multiply by by adding the exponents.
Step 3.22.3.1.2.1.2.1
Move .
Step 3.22.3.1.2.1.2.2
Use the power rule to combine exponents.
Step 3.22.3.1.2.1.2.3
Add and .
Step 3.22.3.1.2.1.3
Move to the left of .
Step 3.22.3.1.2.1.4
Multiply by .
Step 3.22.3.1.2.1.5
Multiply by .
Step 3.22.3.1.2.2
Add and .
Step 3.22.3.1.2.3
Add and .
Step 3.22.3.1.3
Apply the distributive property.
Step 3.22.3.1.4
Multiply by .
Step 3.22.3.1.5
Multiply by .
Step 3.22.3.1.6
Multiply by by adding the exponents.
Step 3.22.3.1.6.1
Move .
Step 3.22.3.1.6.2
Use the power rule to combine exponents.
Step 3.22.3.1.6.3
Add and .
Step 3.22.3.1.7
Multiply by .
Step 3.22.3.1.8
Multiply by .
Step 3.22.3.1.9
Multiply by .
Step 3.22.3.2
Subtract from .
Step 3.22.4
Reorder terms.
Step 3.22.5
Factor out of .
Step 3.22.5.1
Factor out of .
Step 3.22.5.2
Factor out of .
Step 3.22.5.3
Factor out of .
Step 3.22.5.4
Factor out of .
Step 3.22.5.5
Factor out of .
Step 3.22.6
Factor out of .
Step 3.22.7
Factor out of .
Step 3.22.8
Factor out of .
Step 3.22.9
Rewrite as .
Step 3.22.10
Factor out of .
Step 3.22.11
Rewrite as .
Step 3.22.12
Move the negative in front of the fraction.
Step 4
Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Quotient Rule which states that is where and .
Step 4.3
Differentiate.
Step 4.3.1
Multiply the exponents in .
Step 4.3.1.1
Apply the power rule and multiply exponents, .
Step 4.3.1.2
Multiply by .
Step 4.3.2
By the Sum Rule, the derivative of with respect to is .
Step 4.3.3
Differentiate using the Power Rule which states that is where .
Step 4.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.5
Differentiate using the Power Rule which states that is where .
Step 4.3.6
Multiply by .
Step 4.3.7
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.8
Add and .
Step 4.4
Differentiate using the chain rule, which states that is where and .
Step 4.4.1
To apply the Chain Rule, set as .
Step 4.4.2
Differentiate using the Power Rule which states that is where .
Step 4.4.3
Replace all occurrences of with .
Step 4.5
Simplify with factoring out.
Step 4.5.1
Multiply by .
Step 4.5.2
Factor out of .
Step 4.5.2.1
Factor out of .
Step 4.5.2.2
Factor out of .
Step 4.5.2.3
Factor out of .
Step 4.6
Cancel the common factors.
Step 4.6.1
Factor out of .
Step 4.6.2
Cancel the common factor.
Step 4.6.3
Rewrite the expression.
Step 4.7
By the Sum Rule, the derivative of with respect to is .
Step 4.8
Differentiate using the Power Rule which states that is where .
Step 4.9
Since is constant with respect to , the derivative of with respect to is .
Step 4.10
Combine fractions.
Step 4.10.1
Add and .
Step 4.10.2
Multiply by .
Step 4.10.3
Combine and .
Step 4.10.4
Move the negative in front of the fraction.
Step 4.11
Simplify.
Step 4.11.1
Apply the distributive property.
Step 4.11.2
Apply the distributive property.
Step 4.11.3
Apply the distributive property.
Step 4.11.4
Simplify the numerator.
Step 4.11.4.1
Simplify each term.
Step 4.11.4.1.1
Expand using the FOIL Method.
Step 4.11.4.1.1.1
Apply the distributive property.
Step 4.11.4.1.1.2
Apply the distributive property.
Step 4.11.4.1.1.3
Apply the distributive property.
Step 4.11.4.1.2
Simplify and combine like terms.
Step 4.11.4.1.2.1
Simplify each term.
Step 4.11.4.1.2.1.1
Rewrite using the commutative property of multiplication.
Step 4.11.4.1.2.1.2
Multiply by by adding the exponents.
Step 4.11.4.1.2.1.2.1
Move .
Step 4.11.4.1.2.1.2.2
Use the power rule to combine exponents.
Step 4.11.4.1.2.1.2.3
Add and .
Step 4.11.4.1.2.1.3
Rewrite using the commutative property of multiplication.
Step 4.11.4.1.2.1.4
Multiply by by adding the exponents.
Step 4.11.4.1.2.1.4.1
Move .
Step 4.11.4.1.2.1.4.2
Multiply by .
Step 4.11.4.1.2.1.4.2.1
Raise to the power of .
Step 4.11.4.1.2.1.4.2.2
Use the power rule to combine exponents.
Step 4.11.4.1.2.1.4.3
Add and .
Step 4.11.4.1.2.1.5
Multiply by .
Step 4.11.4.1.2.1.6
Multiply by .
Step 4.11.4.1.2.2
Add and .
Step 4.11.4.1.3
Apply the distributive property.
Step 4.11.4.1.4
Simplify.
Step 4.11.4.1.4.1
Multiply by .
Step 4.11.4.1.4.2
Multiply by .
Step 4.11.4.1.4.3
Multiply by .
Step 4.11.4.1.5
Multiply by by adding the exponents.
Step 4.11.4.1.5.1
Move .
Step 4.11.4.1.5.2
Multiply by .
Step 4.11.4.1.5.2.1
Raise to the power of .
Step 4.11.4.1.5.2.2
Use the power rule to combine exponents.
Step 4.11.4.1.5.3
Add and .
Step 4.11.4.1.6
Multiply by .
Step 4.11.4.1.7
Multiply by by adding the exponents.
Step 4.11.4.1.7.1
Move .
Step 4.11.4.1.7.2
Multiply by .
Step 4.11.4.1.7.2.1
Raise to the power of .
Step 4.11.4.1.7.2.2
Use the power rule to combine exponents.
Step 4.11.4.1.7.3
Add and .
Step 4.11.4.1.8
Multiply by .
Step 4.11.4.1.9
Multiply by .
Step 4.11.4.1.10
Multiply by .
Step 4.11.4.1.11
Multiply by .
Step 4.11.4.2
Subtract from .
Step 4.11.4.3
Add and .
Step 4.11.4.4
Subtract from .
Step 4.11.5
Factor out of .
Step 4.11.5.1
Factor out of .
Step 4.11.5.2
Factor out of .
Step 4.11.5.3
Factor out of .
Step 4.11.5.4
Factor out of .
Step 4.11.5.5
Factor out of .
Step 4.11.6
Factor out of .
Step 4.11.7
Factor out of .
Step 4.11.8
Factor out of .
Step 4.11.9
Rewrite as .
Step 4.11.10
Factor out of .
Step 4.11.11
Rewrite as .
Step 4.11.12
Move the negative in front of the fraction.
Step 4.11.13
Multiply by .
Step 4.11.14
Multiply by .
Step 4.11.15
Reorder factors in .
Step 5
The fourth derivative of with respect to is .