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. Identify the outer function as the natural logarithm and the inner function as u=x−√(,x2−1)

2. Apply the chain rule for the natural logarithm, which states d(ln(u))/d(x)=1/u⋅d(u)/d(x)

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

3. Differentiate the inner expression term by term.

d(x−√(,x2−1))/d(x)=1−d(x2−1)/d(x)

4. Apply the chain rule to the square root term.

d(x2−1)/d(x)=1/2*(x2−1)(−1/2)⋅2*x

d(x2−1)/d(x)=x/√(,x2−1)

5. Substitute the derivative of the inner expression back into the chain rule formula.

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

6. Simplify the expression by finding a common denominator for the term in parentheses.

1−x/√(,x2−1)=(√(,x2−1)−x)/√(,x2−1)

7. Combine the fractions and simplify the signs.

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

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

8. Cancel the common factor (x−√(,x2−1))

d(ln(x−√(,x2−1)))/d(x)=(−1)/√(,x2−1)

Final Answer

d(ln(x−√(,x2−1)))/d(x)=−1/√(,x2−1)

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