Calculus Examples

Find the Derivative - d/dx (x^2+3y^2)/(e^(x^2+y^2))
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate.
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Step 2.1
Multiply the exponents in .
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Step 2.1.1
Apply the power rule and multiply exponents, .
Step 2.1.2
Apply the distributive property.
Step 2.1.3
Move to the left of .
Step 2.1.4
Move to the left of .
Step 2.1.5
Multiply by .
Step 2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.5
Add and .
Step 3
Differentiate using the chain rule, which states that is where and .
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Step 3.1
To apply the Chain Rule, set as .
Step 3.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.3
Replace all occurrences of with .
Step 4
Differentiate.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.4
Simplify the expression.
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Step 4.4.1
Add and .
Step 4.4.2
Multiply by .
Step 5
Simplify.
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Step 5.1
Apply the distributive property.
Step 5.2
Simplify the numerator.
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Step 5.2.1
Simplify each term.
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Step 5.2.1.1
Rewrite using the commutative property of multiplication.
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Apply the distributive property.
Step 5.2.1.4
Multiply by by adding the exponents.
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Step 5.2.1.4.1
Move .
Step 5.2.1.4.2
Multiply by .
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Step 5.2.1.4.2.1
Raise to the power of .
Step 5.2.1.4.2.2
Use the power rule to combine exponents.
Step 5.2.1.4.3
Add and .
Step 5.2.2
Reorder factors in .
Step 5.3
Reorder terms.
Step 5.4
Factor out of .
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Step 5.4.1
Factor out of .
Step 5.4.2
Factor out of .
Step 5.4.3
Factor out of .
Step 5.4.4
Factor out of .
Step 5.4.5
Factor out of .
Step 5.5
Cancel the common factor of and .
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Step 5.5.1
Factor out of .
Step 5.5.2
Cancel the common factors.
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Step 5.5.2.1
Multiply by .
Step 5.5.2.2
Cancel the common factor.
Step 5.5.2.3
Rewrite the expression.
Step 5.5.2.4
Divide by .
Step 5.6
Simplify each term.
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Step 5.6.1
Apply the distributive property.
Step 5.6.2
Multiply by .
Step 5.6.3
Multiply by .
Step 5.7
Subtract from .
Step 5.8
Subtract from .
Step 5.9
Apply the distributive property.
Step 5.10
Simplify.
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Step 5.10.1
Multiply by .
Step 5.10.2
Multiply by by adding the exponents.
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Step 5.10.2.1
Move .
Step 5.10.2.2
Multiply by .
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Step 5.10.2.2.1
Raise to the power of .
Step 5.10.2.2.2
Use the power rule to combine exponents.
Step 5.10.2.3
Add and .
Step 5.10.3
Multiply by .
Step 5.11
Multiply by .
Step 5.12
Reorder factors in .