Calculus Examples

Find the Derivative - d/dx (e^x-e^(-x))/(e^x+e^(-x))
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 Exponential Rule which states that is where =.
Step 4
Since is constant with respect to , 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
Multiply.
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Step 6.2.1
Multiply by .
Step 6.2.2
Multiply by .
Step 6.3
Differentiate using the Power Rule which states that is where .
Step 6.4
Multiply by .
Step 7
Raise to the power of .
Step 8
Raise to the power of .
Step 9
Use the power rule to combine exponents.
Step 10
Differentiate using the Sum Rule.
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Step 10.1
Add and .
Step 10.2
By the Sum Rule, the derivative of with respect to is .
Step 11
Differentiate using the Exponential Rule which states that is where =.
Step 12
Differentiate using the chain rule, which states that is where and .
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Step 12.1
To apply the Chain Rule, set as .
Step 12.2
Differentiate using the Exponential Rule which states that is where =.
Step 12.3
Replace all occurrences of with .
Step 13
Differentiate.
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Step 13.1
Since is constant with respect to , the derivative of with respect to is .
Step 13.2
Differentiate using the Power Rule which states that is where .
Step 13.3
Simplify the expression.
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Step 13.3.1
Multiply by .
Step 13.3.2
Move to the left of .
Step 13.3.3
Rewrite as .
Step 14
Raise to the power of .
Step 15
Raise to the power of .
Step 16
Use the power rule to combine exponents.
Step 17
Add and .
Step 18
Simplify the numerator.
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Step 18.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 18.2
Simplify.
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Step 18.2.1
Add and .
Step 18.2.2
Subtract from .
Step 18.2.3
Add and .
Step 18.2.4
Apply the distributive property.
Step 18.2.5
Multiply .
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Step 18.2.5.1
Multiply by .
Step 18.2.5.2
Multiply by .
Step 18.2.6
Subtract from .
Step 18.2.7
Add and .
Step 18.2.8
Add and .
Step 18.2.9
Combine exponents.
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Step 18.2.9.1
Multiply by .
Step 18.2.9.2
Multiply by by adding the exponents.
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Step 18.2.9.2.1
Move .
Step 18.2.9.2.2
Use the power rule to combine exponents.
Step 18.2.9.2.3
Add and .
Step 18.2.9.3
Simplify .