Calculus Examples

Find the Derivative - d/dx natural log of (x+1)/(x-1)
Step 1
Differentiate using the chain rule, which states that is where and .
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Step 1.1
To apply the Chain Rule, set as .
Step 1.2
The derivative of with respect to is .
Step 1.3
Replace all occurrences of with .
Step 2
Multiply by the reciprocal of the fraction to divide by .
Step 3
Multiply by .
Step 4
Differentiate using the Quotient Rule which states that is where and .
Step 5
Differentiate.
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Step 5.1
By the Sum Rule, the derivative of with respect to is .
Step 5.2
Differentiate using the Power Rule which states that is where .
Step 5.3
Since is constant with respect to , the derivative of with respect to is .
Step 5.4
Simplify the expression.
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Step 5.4.1
Add and .
Step 5.4.2
Multiply by .
Step 5.5
By the Sum Rule, the derivative of with respect to is .
Step 5.6
Differentiate using the Power Rule which states that is where .
Step 5.7
Since is constant with respect to , the derivative of with respect to is .
Step 5.8
Combine fractions.
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Step 5.8.1
Add and .
Step 5.8.2
Multiply by .
Step 5.8.3
Multiply by .
Step 6
Cancel the common factors.
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Step 6.1
Factor out of .
Step 6.2
Cancel the common factor.
Step 6.3
Rewrite the expression.
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
Combine the opposite terms in .
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Step 7.2.1.1
Subtract from .
Step 7.2.1.2
Subtract from .
Step 7.2.2
Multiply by .
Step 7.2.3
Subtract from .
Step 7.3
Move the negative in front of the fraction.