Find the Derivative - d/dx

Problem

d()/d(x)*ln(√(,(x−1)/(x+1)))

Solution

1. Rewrite the expression using the properties of logarithms and exponents to simplify the differentiation process.

ln(√(,(x−1)/(x+1)))=ln((x−1)/(x+1))

2. Apply the power rule for logarithms, ln(ab)=b*ln(a) to move the exponent to the front.

1/2*ln((x−1)/(x+1))

3. Apply the quotient rule for logarithms, ln(a/b)=ln(a)−ln(b) to split the expression into two terms.

1/2*(ln(x−1)−ln(x+1))

4. Differentiate each term using the rule d(ln(u))/d(x)=1/ud(u)/d(x)

d()/d(x)*(1/2*(ln(x−1)−ln(x+1)))=1/2*(1/(x−1)−1/(x+1))

5. Find a common denominator to combine the fractions inside the parentheses.

1/2*(((x+1)−(x−1))/((x−1)*(x+1)))

6. Simplify the numerator and the denominator.

1/2*(2/(x2−1))

7. Cancel the common factor of 2.

1/(x2−1)

Final Answer

d()/d(x)*ln(√(,(x−1)/(x+1)))=1/(x2−1)

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