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

Problem 2.1476 · hard

1. \[ \frac{d}{d x} \ln{\left(x + \sqrt{x^{2} + 1} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(x + \sqrt{x^{2} + 1}\right)}{x + \sqrt{x^{2} + 1}} \]
   chainApply the chain rule for the logarithm.✓ Proved
3. \[ = \frac{\frac{d}{d x} x + \frac{d}{d x} \sqrt{x^{2} + 1}}{x + \sqrt{x^{2} + 1}} \]
   sumApply the sum rule to the inner expression.✓ Proved
4. \[ = \frac{\frac{d}{d x} \sqrt{x^{2} + 1} + 1}{x + \sqrt{x^{2} + 1}} \]
   constant rewriteThe derivative of x is 1. Rewrite the square root as a power.✓ Proved
5. \[ = \frac{1 + \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \sqrt{x^{2} + 1}}}{x + \sqrt{x^{2} + 1}} \]
   powerApply the power rule.✓ Proved
6. \[ = \frac{\frac{x}{\sqrt{x^{2} + 1}} + 1}{x + \sqrt{x^{2} + 1}} \]
   derivative constant-multiple rewriteDifferentiate the inner polynomial. Simplify the constant 1/2 * 2. Rewrite the negative power as a reciprocal.✓ Proved
7. \[ = \frac{1}{\sqrt{x^{2} + 1}} \]
   algebra algebra simplifyFind a common denominator for the numerator. Multiply the numerator by the reciprocal. Cancel the common term (x + sqrt(x**2 + 1)).✓ Proved