Find the Derivative - d/dx natural log of x square root of x^2-1

Problem

d(ln(x√(,x2−1)))/d(x)

Solution

1. Apply logarithm properties to simplify the expression before differentiating. Use the property ln(a*b)=ln(a)+ln(b)

ln(x√(,x2−1))=ln(x)+ln(√(,x2−1))

2. Rewrite the square root as an exponent of 1/2 and apply the power rule for logarithms, ln(an)=n*ln(a)

ln(x)+ln((x2−1)(1/2))=ln(x)+1/2*ln(x2−1)

3. Differentiate the first term using the rule d(ln(x))/d(x)=1/x

d(ln(x))/d(x)=1/x

4. Differentiate the second term using the chain rule, where d(ln(u))/d(x)=1/u⋅d(u)/d(x)

d()/d(x)1/2*ln(x2−1)=1/2⋅1/(x2−1)⋅2*x

5. Simplify the derivative of the second term by canceling the constant factors.

1/2⋅(2*x)/(x2−1)=x/(x2−1)

6. Combine the results from both terms into a single expression.

1/x+x/(x2−1)

7. Find a common denominator to simplify the final expression.

(x2−1+x2)/(x*(x2−1))=(2*x2−1)/(x*(x2−1))

Final Answer

d(ln(x√(,x2−1)))/d(x)=(2*x2−1)/(x*(x2−1))

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