Find the Derivative - d/dx natural log of (x+1)/(x-1)

Problem

d(ln((x+1)/(x−1)))/d(x)

Solution

1. Apply logarithm properties to simplify the expression before differentiating by using the rule ln(a/b)=ln(a)−ln(b)

ln((x+1)/(x−1))=ln(x+1)−ln(x−1)

2. Differentiate the terms individually using the chain rule for natural logarithms, where d(ln(u))/d(x)=1/u⋅d(u)/d(x)

d(ln(x+1))/d(x)−d(ln(x−1))/d(x)

3. Calculate the derivatives of the inner functions, noting that d(x+1)/d(x)=1 and d(x−1)/d(x)=1

1/(x+1)−1/(x−1)

4. Find a common denominator to combine the fractions, which is (x+1)*(x−1)=x2−1

((x−1)−(x+1))/((x+1)*(x−1))

5. Simplify the numerator by distributing the negative sign and combining like terms.

(x−1−x−1)/(x2−1)

(−2)/(x2−1)

Final Answer

d(ln((x+1)/(x−1)))/d(x)=(−2)/(x2−1)

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