Calculus Examples

Find the Derivative - d/dx (e^(2x)+1)/(e^(2x)-1)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
By the Sum Rule, the derivative of with respect to is .
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
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Simplify the expression.
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Step 4.3.1
Multiply by .
Step 4.3.2
Move to the left of .
Step 4.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.5
Simplify the expression.
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Step 4.5.1
Add and .
Step 4.5.2
Move to the left of .
Step 4.6
By the Sum Rule, the derivative of with respect to is .
Step 5
Differentiate using the chain rule, which states that is where and .
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Step 5.1
To apply the Chain Rule, set as .
Step 5.2
Differentiate using the Exponential Rule which states that is where =.
Step 5.3
Replace all occurrences of with .
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
Simplify the expression.
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Step 6.3.1
Multiply by .
Step 6.3.2
Move to the left of .
Step 6.4
Since is constant with respect to , the derivative of with respect to is .
Step 6.5
Simplify the expression.
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Step 6.5.1
Add and .
Step 6.5.2
Multiply by .
Step 7
Simplify.
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Step 7.1
Apply the distributive property.
Step 7.2
Apply the distributive property.
Step 7.3
Apply the distributive property.
Step 7.4
Apply the distributive property.
Step 7.5
Simplify the numerator.
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Step 7.5.1
Combine the opposite terms in .
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Step 7.5.1.1
Subtract from .
Step 7.5.1.2
Add and .
Step 7.5.2
Simplify each term.
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Step 7.5.2.1
Multiply by .
Step 7.5.2.2
Multiply by .
Step 7.5.3
Subtract from .
Step 7.6
Move the negative in front of the fraction.
Step 7.7
Simplify the denominator.
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Step 7.7.1
Rewrite as .
Step 7.7.2
Rewrite as .
Step 7.7.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 7.7.4
Apply the product rule to .