Calculus Examples

Find the 2nd Derivative f(x)=x/(x^2+1)
Step 1
Find the first derivative.
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Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
Differentiate.
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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.
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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
Find the second derivative.
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Step 2.1
Differentiate using the Quotient Rule which states that is where and .
Step 2.2
Differentiate.
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Step 2.2.1
Multiply the exponents in .
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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 .
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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.
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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.
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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.
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Step 2.5.1
Apply the distributive property.
Step 2.5.2
Apply the distributive property.
Step 2.5.3
Simplify the numerator.
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Step 2.5.3.1
Simplify each term.
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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.
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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.
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Step 2.5.3.1.4.1
Simplify each term.
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Step 2.5.3.1.4.1.1
Multiply by by adding the exponents.
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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.
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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.
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Step 2.5.3.1.8.1
Multiply by by adding the exponents.
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Step 2.5.3.1.8.1.1
Move .
Step 2.5.3.1.8.1.2
Multiply by .
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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.
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Step 2.5.3.1.8.2.1
Move .
Step 2.5.3.1.8.2.2
Multiply by .
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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.
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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.
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Step 2.5.3.1.10.1
Multiply by by adding the exponents.
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Step 2.5.3.1.10.1.1
Multiply by .
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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.
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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.
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Step 2.5.3.1.12.1
Simplify each term.
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Step 2.5.3.1.12.1.1
Multiply by by adding the exponents.
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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.
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Step 2.5.3.1.12.1.2.1
Move .
Step 2.5.3.1.12.1.2.2
Multiply by .
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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.
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Step 2.5.4.1
Factor out of .
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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.
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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 .
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Step 2.5.5.1
Factor out of .
Step 2.5.5.2
Cancel the common factors.
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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
Find the third derivative.
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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 .
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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.
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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.
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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.
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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 .
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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.
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Step 3.11.1
Multiply by .
Step 3.11.2
Factor out of .
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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.
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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.
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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.
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Step 3.22.1
Apply the distributive property.
Step 3.22.2
Apply the distributive property.
Step 3.22.3
Simplify the numerator.
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Step 3.22.3.1
Simplify each term.
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Step 3.22.3.1.1
Expand using the FOIL Method.
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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.
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Step 3.22.3.1.2.1
Simplify each term.
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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.
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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.
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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 .
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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
Find the fourth derivative.
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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.
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Step 4.3.1
Multiply the exponents in .
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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 .
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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.
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Step 4.5.1
Multiply by .
Step 4.5.2
Factor out of .
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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.
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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.
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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.
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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.
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Step 4.11.4.1
Simplify each term.
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Step 4.11.4.1.1
Expand using the FOIL Method.
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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.
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Step 4.11.4.1.2.1
Simplify each term.
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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.
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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.
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Step 4.11.4.1.2.1.4.1
Move .
Step 4.11.4.1.2.1.4.2
Multiply by .
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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.
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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.
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Step 4.11.4.1.5.1
Move .
Step 4.11.4.1.5.2
Multiply by .
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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.
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Step 4.11.4.1.7.1
Move .
Step 4.11.4.1.7.2
Multiply by .
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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 .
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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 .