Calculus Examples

Find the Derivative - d/dx (3-xe^x)/(x+e^x)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Add and .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 3
Differentiate using the Product Rule which states that is where and .
Step 4
Differentiate using the Exponential Rule which states that is where =.
Step 5
Differentiate.
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Step 5.1
Differentiate using the Power Rule which states that is where .
Step 5.2
Multiply by .
Step 5.3
By the Sum Rule, the derivative of with respect to is .
Step 5.4
Differentiate using the Power Rule which states that is where .
Step 6
Differentiate using the Exponential Rule which states that is where =.
Step 7
Simplify.
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Step 7.1
Apply the distributive property.
Step 7.2
Apply the distributive property.
Step 7.3
Simplify the numerator.
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Step 7.3.1
Simplify each term.
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Step 7.3.1.1
Expand using the FOIL Method.
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Step 7.3.1.1.1
Apply the distributive property.
Step 7.3.1.1.2
Apply the distributive property.
Step 7.3.1.1.3
Apply the distributive property.
Step 7.3.1.2
Simplify each term.
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Step 7.3.1.2.1
Rewrite using the commutative property of multiplication.
Step 7.3.1.2.2
Multiply by by adding the exponents.
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Step 7.3.1.2.2.1
Move .
Step 7.3.1.2.2.2
Multiply by .
Step 7.3.1.2.3
Rewrite using the commutative property of multiplication.
Step 7.3.1.2.4
Rewrite using the commutative property of multiplication.
Step 7.3.1.2.5
Multiply by by adding the exponents.
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Step 7.3.1.2.5.1
Move .
Step 7.3.1.2.5.2
Use the power rule to combine exponents.
Step 7.3.1.2.5.3
Add and .
Step 7.3.1.2.6
Rewrite using the commutative property of multiplication.
Step 7.3.1.2.7
Multiply by by adding the exponents.
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Step 7.3.1.2.7.1
Move .
Step 7.3.1.2.7.2
Use the power rule to combine exponents.
Step 7.3.1.2.7.3
Add and .
Step 7.3.1.3
Simplify each term.
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Step 7.3.1.3.1
Multiply by .
Step 7.3.1.3.2
Multiply .
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Step 7.3.1.3.2.1
Multiply by .
Step 7.3.1.3.2.2
Multiply by .
Step 7.3.1.4
Expand using the FOIL Method.
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Step 7.3.1.4.1
Apply the distributive property.
Step 7.3.1.4.2
Apply the distributive property.
Step 7.3.1.4.3
Apply the distributive property.
Step 7.3.1.5
Simplify each term.
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Step 7.3.1.5.1
Multiply by .
Step 7.3.1.5.2
Multiply by .
Step 7.3.1.5.3
Multiply by by adding the exponents.
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Step 7.3.1.5.3.1
Move .
Step 7.3.1.5.3.2
Use the power rule to combine exponents.
Step 7.3.1.5.3.3
Add and .
Step 7.3.2
Combine the opposite terms in .
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Step 7.3.2.1
Add and .
Step 7.3.2.2
Add and .
Step 7.3.2.3
Reorder the factors in the terms and .
Step 7.3.2.4
Add and .
Step 7.3.2.5
Add and .
Step 7.4
Reorder terms.
Step 7.5
Factor out of .
Step 7.6
Rewrite as .
Step 7.7
Factor out of .
Step 7.8
Factor out of .
Step 7.9
Factor out of .
Step 7.10
Factor out of .
Step 7.11
Factor out of .
Step 7.12
Rewrite as .
Step 7.13
Move the negative in front of the fraction.
Step 7.14
Reorder factors in .