Calculus Examples

Find the Derivative - d/dx y=(x^36 square root of 29x-4)/((x-1)^14)
Step 1
Use to rewrite as .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Multiply the exponents in .
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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
Differentiate using the chain rule, which states that is where and .
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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
Simplify the numerator.
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Step 9.1
Multiply by .
Step 9.2
Subtract from .
Step 10
Combine fractions.
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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
Combine fractions.
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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
Combine and using a common denominator.
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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
Multiply by by adding the exponents.
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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
Cancel the common factor of .
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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
Simplify the expression.
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Step 30.1
Combine the numerators over the common denominator.
Step 30.2
Add and .
Step 31
Cancel the common factor of .
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Step 31.1
Cancel the common factor.
Step 31.2
Rewrite the expression.
Step 32
Simplify.
Step 33
Differentiate using the chain rule, which states that is where and .
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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
Differentiate.
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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.
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Step 34.5.1
Add and .
Step 34.5.2
Multiply by .
Step 35
Simplify.
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Step 35.1
Apply the distributive property.
Step 35.2
Apply the distributive property.
Step 35.3
Simplify the numerator.
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Step 35.3.1
Factor out of .
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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.
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Step 35.3.2.1
Rewrite using the commutative property of multiplication.
Step 35.3.2.2
Multiply by by adding the exponents.
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Step 35.3.2.2.1
Move .
Step 35.3.2.2.2
Multiply by .
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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.
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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.
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Step 35.3.5.1
Simplify each term.
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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.
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Step 35.3.5.1.2.1
Move .
Step 35.3.5.1.2.2
Multiply by .
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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.
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Step 35.3.5.1.4.1
Move .
Step 35.3.5.1.4.2
Multiply by .
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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.
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Step 35.3.7.1
Move .
Step 35.3.7.2
Multiply by .
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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 .
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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.
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Step 35.4.1
Factor out of .
Step 35.4.2
Cancel the common factors.
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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.