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

Problem 2.1485 · hard Beautiful

1. \[ \frac{d}{d x} \left(x \operatorname{atan}{\left(x \right)} - \frac{\ln{\left(x^{2} + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x \operatorname{atan}{\left(x \right)} - \frac{d}{d x} \frac{\ln{\left(x^{2} + 1 \right)}}{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} x \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   constantPull out the constant factor.✓ Proved
4. \[ = x \frac{d}{d x} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)} \frac{d}{d x} x - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} \operatorname{atan}{\left(x \right)} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   derivative simplifyDifferentiate x. Simplify the expression.✓ Proved
6. \[ = \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \ln{\left(x^{2} + 1 \right)}}{2} \]
   inverse-trig algebraDifferentiate the arctangent function. Simplify the fraction.✓ Proved
7. \[ = \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} - \frac{\frac{d}{d x} \left(x^{2} + 1\right)}{2 \left(x^{2} + 1\right)} \]
   chainApply the chain rule to the logarithm.✓ Proved
8. \[ = \operatorname{atan}{\left(x \right)} \]
   derivative algebra algebra simplifyDifferentiate the inner function x**2 + 1. Simplify the product. Combine the fractions. Simplify the final result.✓ Proved