Derivative of \( \displaystyle - x + \ln{\left(\left(3 x\right)^{x} \right)} \)

Problem 2.2107 · hard

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