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

Problem 2.724 · hard Beautiful

1. \[ \frac{d}{d x} \ln{\left(2 \sqrt{x^{2} + 1} \right)} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(\ln{\left(\sqrt{x^{2} + 1} \right)} + \ln{\left(2 \right)}\right) \]
   algebra algebraUse the logarithm product rule to split the terms. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{d}{d x} \ln{\left(2 \right)} + \frac{d}{d x} \ln{\left(\sqrt{x^{2} + 1} \right)} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = \frac{d}{d x} \ln{\left(\sqrt{x^{2} + 1} \right)} \]
   constant rewriteThe derivative of the constant log(2) is zero. Rewrite the power using the exponential and logarithm identity.✓ Proved
5. \[ = \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
   chainApply the chain rule to the composition of functions.✓ Proved
6. \[ = \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   constant-multiplePull out the constant factor.✓ Proved
7. \[ = \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} x^{2}}{2 \left(x^{2} + 1\right)} \]
   sumApply the sum rule to the inner function.✓ Proved
9. \[ = \frac{x}{x^{2} + 1} \]
   derivative constant simplifyDifferentiate the terms inside the parentheses. The derivative of the constant 1 is zero. Simplify the final expression by canceling the 2.✓ Proved