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Calculus Examples
Step 1
Use to rewrite as .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Multiply by .
Step 4
Differentiate using the Product Rule which states that is where and .
Step 5
Step 5.1
To apply the Chain Rule, set as .
Step 5.2
Differentiate using the Power Rule which states that is where .
Step 5.3
Replace all occurrences of with .
Step 6
To write as a fraction with a common denominator, multiply by .
Step 7
Combine and .
Step 8
Combine the numerators over the common denominator.
Step 9
Step 9.1
Multiply by .
Step 9.2
Subtract from .
Step 10
Step 10.1
Move the negative in front of the fraction.
Step 10.2
Combine and .
Step 10.3
Move to the denominator using the negative exponent rule .
Step 10.4
Combine and .
Step 11
By the Sum Rule, the derivative of with respect to is .
Step 12
Since is constant with respect to , the derivative of with respect to is .
Step 13
Differentiate using the Power Rule which states that is where .
Step 14
Multiply by .
Step 15
Since is constant with respect to , the derivative of with respect to is .
Step 16
Step 16.1
Add and .
Step 16.2
Combine and .
Step 16.3
Move to the left of .
Step 17
Differentiate using the Power Rule which states that is where .
Step 18
Move to the left of .
Step 19
Step 19.1
Move .
Step 19.2
To write as a fraction with a common denominator, multiply by .
Step 19.3
Combine and .
Step 19.4
Combine the numerators over the common denominator.
Step 20
Multiply by .
Step 21
Step 21.1
Move .
Step 21.2
Use the power rule to combine exponents.
Step 21.3
Combine the numerators over the common denominator.
Step 21.4
Add and .
Step 21.5
Divide by .
Step 22
Simplify .
Step 23
Combine and .
Step 24
Multiply by .
Step 25
Combine.
Step 26
Apply the distributive property.
Step 27
Step 27.1
Cancel the common factor.
Step 27.2
Rewrite the expression.
Step 28
Multiply by .
Step 29
Use the power rule to combine exponents.
Step 30
Step 30.1
Combine the numerators over the common denominator.
Step 30.2
Add and .
Step 31
Step 31.1
Cancel the common factor.
Step 31.2
Rewrite the expression.
Step 32
Simplify.
Step 33
Step 33.1
To apply the Chain Rule, set as .
Step 33.2
Differentiate using the Power Rule which states that is where .
Step 33.3
Replace all occurrences of with .
Step 34
Step 34.1
Multiply by .
Step 34.2
By the Sum Rule, the derivative of with respect to is .
Step 34.3
Differentiate using the Power Rule which states that is where .
Step 34.4
Since is constant with respect to , the derivative of with respect to is .
Step 34.5
Simplify the expression.
Step 34.5.1
Add and .
Step 34.5.2
Multiply by .
Step 35
Step 35.1
Apply the distributive property.
Step 35.2
Apply the distributive property.
Step 35.3
Simplify the numerator.
Step 35.3.1
Factor out of .
Step 35.3.1.1
Factor out of .
Step 35.3.1.2
Factor out of .
Step 35.3.1.3
Factor out of .
Step 35.3.2
Simplify each term.
Step 35.3.2.1
Rewrite using the commutative property of multiplication.
Step 35.3.2.2
Multiply by by adding the exponents.
Step 35.3.2.2.1
Move .
Step 35.3.2.2.2
Multiply by .
Step 35.3.2.2.2.1
Raise to the power of .
Step 35.3.2.2.2.2
Use the power rule to combine exponents.
Step 35.3.2.2.3
Add and .
Step 35.3.2.3
Multiply by .
Step 35.3.2.4
Multiply by .
Step 35.3.3
Add and .
Step 35.3.4
Expand using the FOIL Method.
Step 35.3.4.1
Apply the distributive property.
Step 35.3.4.2
Apply the distributive property.
Step 35.3.4.3
Apply the distributive property.
Step 35.3.5
Simplify and combine like terms.
Step 35.3.5.1
Simplify each term.
Step 35.3.5.1.1
Rewrite using the commutative property of multiplication.
Step 35.3.5.1.2
Multiply by by adding the exponents.
Step 35.3.5.1.2.1
Move .
Step 35.3.5.1.2.2
Multiply by .
Step 35.3.5.1.2.2.1
Raise to the power of .
Step 35.3.5.1.2.2.2
Use the power rule to combine exponents.
Step 35.3.5.1.2.3
Add and .
Step 35.3.5.1.3
Rewrite using the commutative property of multiplication.
Step 35.3.5.1.4
Multiply by by adding the exponents.
Step 35.3.5.1.4.1
Move .
Step 35.3.5.1.4.2
Multiply by .
Step 35.3.5.1.4.2.1
Raise to the power of .
Step 35.3.5.1.4.2.2
Use the power rule to combine exponents.
Step 35.3.5.1.4.3
Add and .
Step 35.3.5.1.5
Multiply by .
Step 35.3.5.1.6
Multiply by .
Step 35.3.5.2
Subtract from .
Step 35.3.6
Rewrite using the commutative property of multiplication.
Step 35.3.7
Multiply by by adding the exponents.
Step 35.3.7.1
Move .
Step 35.3.7.2
Multiply by .
Step 35.3.7.2.1
Raise to the power of .
Step 35.3.7.2.2
Use the power rule to combine exponents.
Step 35.3.7.3
Add and .
Step 35.3.8
Multiply by .
Step 35.3.9
Multiply by .
Step 35.3.10
Subtract from .
Step 35.3.11
Add and .
Step 35.3.12
Factor out of .
Step 35.3.12.1
Factor out of .
Step 35.3.12.2
Factor out of .
Step 35.3.12.3
Factor out of .
Step 35.3.12.4
Factor out of .
Step 35.3.12.5
Factor out of .
Step 35.4
Combine terms.
Step 35.4.1
Factor out of .
Step 35.4.2
Cancel the common factors.
Step 35.4.2.1
Factor out of .
Step 35.4.2.2
Cancel the common factor.
Step 35.4.2.3
Rewrite the expression.