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

Problem 2.1260 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(\frac{x}{\sqrt{x^{2} + 1}} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(\ln{\left(x \right)} - \ln{\left(\sqrt{x^{2} + 1} \right)}\right) \]
   algebra algebraUse logarithm properties to split the fraction. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \ln{\left(\sqrt{x^{2} + 1} \right)} \]
   sumApply the difference rule for derivatives.✓ Proved
4. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
   constant-multiplePull out the constant factor 1/2 using the power rule for logs.✓ Proved
5. \[ = \frac{d}{d x} \ln{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   constant-multipleApply the constant multiple rule.✓ Proved
6. \[ = - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} + \frac{1}{x} \]
   derivativeDifferentiate the first term.✓ Proved
7. \[ = - \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} + \frac{1}{x} \]
   chainApply the chain rule to the second term.✓ Proved
8. \[ = - \frac{x}{x^{2} + 1} + \frac{1}{x} \]
   derivative algebraDifferentiate the inner function x**2 + 1. Simplify the expression by canceling the 2 and the 1/2.✓ Proved
9. \[ = \frac{1}{x \left(x^{2} + 1\right)} \]
   algebra simplifyFind a common denominator to combine the terms. Simplify the numerator.✓ Proved