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

Problem 2.1598 · hard

1. \[ \frac{d}{d x} \left(- x + \ln{\left(\left(2 x\right)^{x} \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \ln{\left(\left(2 x\right)^{x} \right)} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} \ln{\left(e^{x \ln{\left(2 x \right)}} \right)} \]
   rewriteRewrite the power using the exponential and logarithm identity.✓ Proved
4. \[ = \frac{d}{d x} \left(- x\right) + \frac{d}{d x} x \ln{\left(2 x \right)} \]
   simplifySimplify the log of the exponential.✓ Proved
5. \[ = x \frac{d}{d x} \ln{\left(2 x \right)} + \ln{\left(2 x \right)} \frac{d}{d x} x + \frac{d}{d x} \left(- x\right) \]
   productApply the product rule to the term inside the derivative.✓ Proved
6. \[ = x \frac{d}{d x} \ln{\left(2 x \right)} + \ln{\left(2 x \right)} + \frac{d}{d x} \left(- x\right) \]
   derivativeDifferentiate x.✓ Proved
7. \[ = \ln{\left(2 x \right)} + \frac{d}{d x} \left(- x\right) + \frac{\frac{d}{d x} 2 x}{2} \]
   chainApply the chain rule to log(2*x).✓ Proved
8. \[ = \ln{\left(2 x \right)} + \frac{d}{d x} \left(- x\right) + 1 \]
   derivative constant-multiple algebraDifferentiate 2*x. Simplify the constant multiplier. Simplify the expression inside the parentheses.✓ Proved
9. \[ = \ln{\left(2 x \right)} \]
   derivative algebraDifferentiate -x and simplify. Combine the constant terms.✓ Proved