Calculus Examples

Find the Derivative - d/dx h(x)=(x+x^-1)^3
Step 1
Differentiate using the chain rule, which states that is where and .
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Step 1.1
To apply the Chain Rule, set as .
Step 1.2
Differentiate using the Power Rule which states that is where .
Step 1.3
Replace all occurrences of with .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Differentiate using the Power Rule which states that is where .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 3
Simplify.
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Step 3.1
Rewrite the expression using the negative exponent rule .
Step 3.2
Rewrite the expression using the negative exponent rule .
Step 3.3
Rewrite as .
Step 3.4
Expand using the FOIL Method.
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Step 3.4.1
Apply the distributive property.
Step 3.4.2
Apply the distributive property.
Step 3.4.3
Apply the distributive property.
Step 3.5
Simplify and combine like terms.
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Step 3.5.1
Simplify each term.
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Step 3.5.1.1
Multiply by .
Step 3.5.1.2
Cancel the common factor of .
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Step 3.5.1.2.1
Cancel the common factor.
Step 3.5.1.2.2
Rewrite the expression.
Step 3.5.1.3
Cancel the common factor of .
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Step 3.5.1.3.1
Cancel the common factor.
Step 3.5.1.3.2
Rewrite the expression.
Step 3.5.1.4
Multiply .
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Step 3.5.1.4.1
Multiply by .
Step 3.5.1.4.2
Raise to the power of .
Step 3.5.1.4.3
Raise to the power of .
Step 3.5.1.4.4
Use the power rule to combine exponents.
Step 3.5.1.4.5
Add and .
Step 3.5.2
Add and .
Step 3.6
Apply the distributive property.
Step 3.7
Simplify.
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Step 3.7.1
Multiply by .
Step 3.7.2
Combine and .
Step 3.8
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.9
Simplify each term.
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Step 3.9.1
Multiply by .
Step 3.9.2
Cancel the common factor of .
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Step 3.9.2.1
Move the leading negative in into the numerator.
Step 3.9.2.2
Factor out of .
Step 3.9.2.3
Cancel the common factor.
Step 3.9.2.4
Rewrite the expression.
Step 3.9.3
Multiply by .
Step 3.9.4
Multiply by .
Step 3.9.5
Multiply .
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Step 3.9.5.1
Multiply by .
Step 3.9.5.2
Combine and .
Step 3.9.6
Move the negative in front of the fraction.
Step 3.9.7
Multiply by .
Step 3.9.8
Rewrite using the commutative property of multiplication.
Step 3.9.9
Multiply .
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Step 3.9.9.1
Multiply by .
Step 3.9.9.2
Multiply by by adding the exponents.
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Step 3.9.9.2.1
Use the power rule to combine exponents.
Step 3.9.9.2.2
Add and .
Step 3.10
Combine the numerators over the common denominator.
Step 3.11
Add and .
Step 3.12
Simplify each term.
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Step 3.12.1
Move the negative in front of the fraction.
Step 3.12.2
Move the negative in front of the fraction.
Step 3.13
Add and .