Calculus Examples

Find the Derivative - d/dx f(x) = natural log of ( square root of x^2+1)/(x(2x^3-1)^2)
Step 1
Use to rewrite as .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
The derivative of with respect to is .
Step 2.3
Replace all occurrences of with .
Step 3
Multiply by the reciprocal of the fraction to divide by .
Step 4
Multiply by .
Step 5
Differentiate using the Quotient Rule which states that is where and .
Step 6
Differentiate using the chain rule, which states that is where and .
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Step 6.1
To apply the Chain Rule, set as .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Replace all occurrences of with .
Step 7
To write as a fraction with a common denominator, multiply by .
Step 8
Combine and .
Step 9
Combine the numerators over the common denominator.
Step 10
Simplify the numerator.
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Step 10.1
Multiply by .
Step 10.2
Subtract from .
Step 11
Combine fractions.
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Step 11.1
Move the negative in front of the fraction.
Step 11.2
Combine and .
Step 11.3
Move to the denominator using the negative exponent rule .
Step 11.4
Combine and .
Step 11.5
Combine and .
Step 12
By the Sum Rule, the derivative of with respect to is .
Step 13
Differentiate using the Power Rule which states that is where .
Step 14
Since is constant with respect to , the derivative of with respect to is .
Step 15
Combine fractions.
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Step 15.1
Add and .
Step 15.2
Combine and .
Step 15.3
Combine and .
Step 16
Raise to the power of .
Step 17
Raise to the power of .
Step 18
Use the power rule to combine exponents.
Step 19
Add and .
Step 20
Cancel the common factor.
Step 21
Rewrite the expression.
Step 22
Multiply by .
Step 23
Combine.
Step 24
Apply the distributive property.
Step 25
Cancel the common factor of .
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Step 25.1
Cancel the common factor.
Step 25.2
Rewrite the expression.
Step 26
Multiply by by adding the exponents.
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Step 26.1
Move .
Step 26.2
Use the power rule to combine exponents.
Step 26.3
Combine the numerators over the common denominator.
Step 26.4
Add and .
Step 26.5
Divide by .
Step 27
Simplify .
Step 28
Differentiate using the Product Rule which states that is where and .
Step 29
Differentiate using the chain rule, which states that is where and .
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Step 29.1
To apply the Chain Rule, set as .
Step 29.2
Differentiate using the Power Rule which states that is where .
Step 29.3
Replace all occurrences of with .
Step 30
Differentiate.
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Step 30.1
By the Sum Rule, the derivative of with respect to is .
Step 30.2
Since is constant with respect to , the derivative of with respect to is .
Step 30.3
Differentiate using the Power Rule which states that is where .
Step 30.4
Multiply by .
Step 30.5
Since is constant with respect to , the derivative of with respect to is .
Step 30.6
Simplify the expression.
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Step 30.6.1
Add and .
Step 30.6.2
Multiply by .
Step 31
Raise to the power of .
Step 32
Use the power rule to combine exponents.
Step 33
Add and .
Step 34
Differentiate using the Power Rule which states that is where .
Step 35
Combine fractions.
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Step 35.1
Multiply by .
Step 35.2
Multiply by .
Step 36
Multiply by by adding the exponents.
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Step 36.1
Move .
Step 36.2
Use the power rule to combine exponents.
Step 36.3
Combine the numerators over the common denominator.
Step 36.4
Add and .
Step 36.5
Divide by .
Step 37
Simplify .
Step 38
Simplify.
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Step 38.1
Apply the product rule to .
Step 38.2
Apply the distributive property.
Step 38.3
Apply the distributive property.
Step 38.4
Apply the distributive property.
Step 38.5
Simplify the numerator.
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Step 38.5.1
Combine exponents.
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Step 38.5.1.1
Multiply by .
Step 38.5.1.2
Multiply by .
Step 38.5.1.3
Multiply by by adding the exponents.
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Step 38.5.1.3.1
Move .
Step 38.5.1.3.2
Use the power rule to combine exponents.
Step 38.5.1.3.3
Add and .
Step 38.5.1.4
Multiply by .
Step 38.5.1.5
Multiply by .
Step 38.5.2
Simplify each term.
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Step 38.5.2.1
Rewrite as .
Step 38.5.2.2
Expand using the FOIL Method.
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Step 38.5.2.2.1
Apply the distributive property.
Step 38.5.2.2.2
Apply the distributive property.
Step 38.5.2.2.3
Apply the distributive property.
Step 38.5.2.3
Simplify and combine like terms.
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Step 38.5.2.3.1
Simplify each term.
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Step 38.5.2.3.1.1
Rewrite using the commutative property of multiplication.
Step 38.5.2.3.1.2
Multiply by by adding the exponents.
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Step 38.5.2.3.1.2.1
Move .
Step 38.5.2.3.1.2.2
Use the power rule to combine exponents.
Step 38.5.2.3.1.2.3
Add and .
Step 38.5.2.3.1.3
Multiply by .
Step 38.5.2.3.1.4
Multiply by .
Step 38.5.2.3.1.5
Multiply by .
Step 38.5.2.3.1.6
Multiply by .
Step 38.5.2.3.2
Subtract from .
Step 38.5.2.4
Apply the distributive property.
Step 38.5.2.5
Simplify.
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Step 38.5.2.5.1
Rewrite using the commutative property of multiplication.
Step 38.5.2.5.2
Rewrite using the commutative property of multiplication.
Step 38.5.2.5.3
Multiply by .
Step 38.5.2.6
Simplify each term.
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Step 38.5.2.6.1
Multiply by by adding the exponents.
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Step 38.5.2.6.1.1
Move .
Step 38.5.2.6.1.2
Use the power rule to combine exponents.
Step 38.5.2.6.1.3
Add and .
Step 38.5.2.6.2
Multiply by by adding the exponents.
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Step 38.5.2.6.2.1
Move .
Step 38.5.2.6.2.2
Use the power rule to combine exponents.
Step 38.5.2.6.2.3
Add and .
Step 38.5.2.7
Simplify each term.
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Step 38.5.2.7.1
Rewrite as .
Step 38.5.2.7.2
Expand using the FOIL Method.
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Step 38.5.2.7.2.1
Apply the distributive property.
Step 38.5.2.7.2.2
Apply the distributive property.
Step 38.5.2.7.2.3
Apply the distributive property.
Step 38.5.2.7.3
Simplify and combine like terms.
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Step 38.5.2.7.3.1
Simplify each term.
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Step 38.5.2.7.3.1.1
Rewrite using the commutative property of multiplication.
Step 38.5.2.7.3.1.2
Multiply by by adding the exponents.
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Step 38.5.2.7.3.1.2.1
Move .
Step 38.5.2.7.3.1.2.2
Use the power rule to combine exponents.
Step 38.5.2.7.3.1.2.3
Add and .
Step 38.5.2.7.3.1.3
Multiply by .
Step 38.5.2.7.3.1.4
Multiply by .
Step 38.5.2.7.3.1.5
Multiply by .
Step 38.5.2.7.3.1.6
Multiply by .
Step 38.5.2.7.3.2
Subtract from .
Step 38.5.2.7.4
Apply the distributive property.
Step 38.5.2.7.5
Simplify.
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Step 38.5.2.7.5.1
Multiply by .
Step 38.5.2.7.5.2
Multiply by .
Step 38.5.2.7.5.3
Multiply by .
Step 38.5.2.8
Subtract from .
Step 38.5.2.9
Add and .
Step 38.5.2.10
Expand by multiplying each term in the first expression by each term in the second expression.
Step 38.5.2.11
Simplify each term.
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Step 38.5.2.11.1
Rewrite using the commutative property of multiplication.
Step 38.5.2.11.2
Multiply by by adding the exponents.
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Step 38.5.2.11.2.1
Move .
Step 38.5.2.11.2.2
Use the power rule to combine exponents.
Step 38.5.2.11.2.3
Add and .
Step 38.5.2.11.3
Rewrite using the commutative property of multiplication.
Step 38.5.2.11.4
Multiply by by adding the exponents.
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Step 38.5.2.11.4.1
Move .
Step 38.5.2.11.4.2
Use the power rule to combine exponents.
Step 38.5.2.11.4.3
Add and .
Step 38.5.2.11.5
Move to the left of .
Step 38.5.2.11.6
Rewrite as .
Step 38.5.2.11.7
Multiply by .
Step 38.5.2.11.8
Multiply by .
Step 38.5.2.11.9
Multiply by .
Step 38.5.3
Combine the opposite terms in .
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Step 38.5.3.1
Subtract from .
Step 38.5.3.2
Add and .
Step 38.5.4
Subtract from .
Step 38.5.5
Add and .
Step 38.5.6
Reorder terms.
Step 38.6
Combine terms.
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Step 38.6.1
Multiply the exponents in .
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Step 38.6.1.1
Apply the power rule and multiply exponents, .
Step 38.6.1.2
Multiply by .
Step 38.6.2
Cancel the common factor of and .
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Step 38.6.2.1
Factor out of .
Step 38.6.2.2
Cancel the common factors.
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Step 38.6.2.2.1
Factor out of .
Step 38.6.2.2.2
Cancel the common factor.
Step 38.6.2.2.3
Rewrite the expression.
Step 38.6.3
Cancel the common factors.
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Step 38.6.3.1
Factor out of .
Step 38.6.3.2
Cancel the common factor.
Step 38.6.3.3
Rewrite the expression.
Step 38.7
Reorder terms.
Step 38.8
Factor out of .
Step 38.9
Factor out of .
Step 38.10
Factor out of .
Step 38.11
Factor out of .
Step 38.12
Factor out of .
Step 38.13
Factor out of .
Step 38.14
Factor out of .
Step 38.15
Rewrite as .
Step 38.16
Factor out of .
Step 38.17
Rewrite as .
Step 38.18
Move the negative in front of the fraction.