Calculus Examples

Find the Derivative - d/dx y=(x^2+3e^x)/(2e^x-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
Differentiate using the Power Rule which states that is where .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3
Differentiate using the Exponential Rule which states that is where =.
Step 4
Differentiate.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Since is constant with respect to , the derivative of with respect to is .
Step 5
Differentiate using the Exponential Rule which states that is where =.
Step 6
Differentiate.
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Step 6.1
Since is constant with respect to , the derivative of with respect to is .
Step 6.2
Differentiate using the Power Rule which states that is where .
Step 6.3
Multiply by .
Step 7
Simplify.
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Step 7.1
Apply the distributive property.
Step 7.2
Simplify the numerator.
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Step 7.2.1
Simplify each term.
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Step 7.2.1.1
Expand using the FOIL Method.
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Step 7.2.1.1.1
Apply the distributive property.
Step 7.2.1.1.2
Apply the distributive property.
Step 7.2.1.1.3
Apply the distributive property.
Step 7.2.1.2
Simplify and combine like terms.
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Step 7.2.1.2.1
Simplify each term.
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Step 7.2.1.2.1.1
Rewrite using the commutative property of multiplication.
Step 7.2.1.2.1.2
Multiply by .
Step 7.2.1.2.1.3
Rewrite using the commutative property of multiplication.
Step 7.2.1.2.1.4
Multiply by by adding the exponents.
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Step 7.2.1.2.1.4.1
Move .
Step 7.2.1.2.1.4.2
Use the power rule to combine exponents.
Step 7.2.1.2.1.4.3
Add and .
Step 7.2.1.2.1.5
Multiply by .
Step 7.2.1.2.1.6
Rewrite using the commutative property of multiplication.
Step 7.2.1.2.1.7
Multiply by by adding the exponents.
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Step 7.2.1.2.1.7.1
Move .
Step 7.2.1.2.1.7.2
Multiply by .
Step 7.2.1.2.1.8
Multiply by .
Step 7.2.1.2.1.9
Rewrite using the commutative property of multiplication.
Step 7.2.1.2.1.10
Multiply by .
Step 7.2.1.2.2
Subtract from .
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Step 7.2.1.2.2.1
Move .
Step 7.2.1.2.2.2
Subtract from .
Step 7.2.1.3
Multiply by .
Step 7.2.1.4
Multiply by .
Step 7.2.1.5
Expand using the FOIL Method.
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Step 7.2.1.5.1
Apply the distributive property.
Step 7.2.1.5.2
Apply the distributive property.
Step 7.2.1.5.3
Apply the distributive property.
Step 7.2.1.6
Simplify each term.
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Step 7.2.1.6.1
Rewrite using the commutative property of multiplication.
Step 7.2.1.6.2
Multiply by .
Step 7.2.1.6.3
Multiply .
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Step 7.2.1.6.3.1
Multiply by .
Step 7.2.1.6.3.2
Multiply by .
Step 7.2.1.6.4
Rewrite using the commutative property of multiplication.
Step 7.2.1.6.5
Multiply by by adding the exponents.
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Step 7.2.1.6.5.1
Move .
Step 7.2.1.6.5.2
Use the power rule to combine exponents.
Step 7.2.1.6.5.3
Add and .
Step 7.2.1.6.6
Multiply by .
Step 7.2.1.6.7
Multiply by .
Step 7.2.2
Combine the opposite terms in .
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Step 7.2.2.1
Subtract from .
Step 7.2.2.2
Add and .
Step 7.2.3
Add and .
Step 7.3
Reorder terms.
Step 7.4
Factor out of .
Step 7.5
Factor out of .
Step 7.6
Factor out of .
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
Rewrite as .
Step 7.12
Move the negative in front of the fraction.
Step 7.13
Reorder factors in .