Derivative of \( \displaystyle - x + \ln{\left(\frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \right)} \)

Problem 2.533 · hard Beautiful

1. \[ \frac{d}{d x} \left(- x + \ln{\left(\frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \ln{\left(\frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \right)} \]
   sum algebraApply the sum rule. Rewrite the quotient as a product using exponents.✓ Proved
3. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \frac{\sqrt{x^{2} - 1}}{\sqrt{1 - x^{2}}} \]
   rewrite algebra algebraRewrite the log argument using the exponential identity. Apply the logarithm product rule. Apply the power rule for logarithms.✓ Proved
4. \[ = \frac{d}{d x} \left(- x\right) + \frac{\sqrt{x^{2} - 1} \frac{d}{d x} \left(- \frac{\ln{\left(1 - x^{2} \right)}}{2} + \frac{\ln{\left(x^{2} - 1 \right)}}{2}\right)}{\sqrt{1 - x^{2}}} \]
   chainApply the chain rule.✓ Proved
5. \[ = \frac{d}{d x} \left(- x\right) + \frac{\sqrt{x^{2} - 1} \left(- \frac{\frac{d}{d x} \ln{\left(1 - x^{2} \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(x^{2} - 1 \right)}}{2}\right)}{\sqrt{1 - x^{2}}} \]
   constant-multipleDistribute the derivative across the subtraction.✓ Proved
6. \[ = \frac{d}{d x} \left(- x\right) + \frac{\sqrt{x^{2} - 1} \left(\frac{x}{x^{2} - 1} + \frac{x}{1 - x^{2}}\right)}{\sqrt{1 - x^{2}}} \]
   chain algebraApply the chain rule to the logarithmic terms. Simplify the fractions.✓ Proved
7. \[ = \frac{d}{d x} \left(- x\right) \]
   algebra algebraRewrite 1/(1 - x**2) as -1/(x**2 - 1). Combine the terms in the parentheses.✓ Proved
8. \[ = -1 \]
   algebra simplifyEvaluate the derivative of -x and the zero term. Final simplification.✓ Proved