Calculus Examples

Find the Derivative - d/dx y=(e^(2x))/(1+e^(-2x))
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate.
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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 Power Rule which states that is where .
Step 3.3
Simplify the expression.
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Step 3.3.1
Multiply by .
Step 3.3.2
Move to the left of .
Step 3.4
By the Sum Rule, the derivative of with respect to is .
Step 3.5
Since is constant with respect to , the derivative of with respect to is .
Step 3.6
Add and .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.3
Replace all occurrences of with .
Step 5
Use the power rule to combine exponents.
Step 6
Simplify by adding terms.
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Step 6.1
Add and .
Step 6.2
Simplify the expression.
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Step 6.2.1
Anything raised to is .
Step 6.2.2
Multiply by .
Step 7
Since is constant with respect to , the derivative of with respect to is .
Step 8
Multiply by .
Step 9
Differentiate using the Power Rule which states that is where .
Step 10
Multiply by .
Step 11
Simplify.
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Step 11.1
Apply the distributive property.
Step 11.2
Apply the distributive property.
Step 11.3
Simplify the numerator.
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Step 11.3.1
Simplify each term.
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Step 11.3.1.1
Multiply by .
Step 11.3.1.2
Multiply by by adding the exponents.
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Step 11.3.1.2.1
Move .
Step 11.3.1.2.2
Use the power rule to combine exponents.
Step 11.3.1.2.3
Subtract from .
Step 11.3.1.3
Simplify .
Step 11.3.2
Add and .
Step 11.4
Factor out of .
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Step 11.4.1
Factor out of .
Step 11.4.2
Factor out of .
Step 11.4.3
Factor out of .