Derivative of \( \displaystyle \frac{\ln{\left(\cos{\left(5 x \right)} \right)}}{5} \)

Problem 2.272 · hard

1. \[ \frac{d}{d x} \frac{\ln{\left(\cos{\left(5 x \right)} \right)}}{5} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} \right)}}{5} \]
   constant-multiplePull out the constant factor 1/5.✓ Proved
3. \[ = \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{5 \cos{\left(5 x \right)}} \]
   chainApply the chain rule to the logarithm.✓ Proved
4. \[ = - \frac{\sin{\left(5 x \right)} \frac{d}{d x} 5 x}{5 \cos{\left(5 x \right)}} \]
   chainApply the chain rule to the cosine function.✓ Proved
5. \[ = - \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} \]
   derivative algebraDifferentiate the innermost function 5*x. Simplify the expression by canceling the 5 and 1/5.✓ Proved
6. \[ = - \tan{\left(5 x \right)} \]
   simplifyUse the trigonometric identity sin(u)/cos(u) = tan(u).✓ Proved