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

Problem 2.56 · hard

1. \[ \frac{d}{d x} \left(- \ln{\left(\cos{\left(5 x \right)} \right)}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} \ln{\left(\cos{\left(5 x \right)} \right)} \]
   constantPull out the negative sign.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \cos{\left(5 x \right)}}{\cos{\left(5 x \right)}} \]
   logarithmicApply the derivative rule for the natural logarithm.✓ Proved
4. \[ = \frac{\sin{\left(5 x \right)} \frac{d}{d x} 5 x}{\cos{\left(5 x \right)}} \]
   chainApply the chain rule to the cosine function.✓ Proved
5. \[ = \frac{5 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} \]
   derivative algebraDifferentiate the innermost function 5*x. Simplify the signs and multiply the terms.✓ Proved
6. \[ = 5 \tan{\left(5 x \right)} \]
   simplifyUse the trigonometric identity tan(u) = sin(u)/cos(u).✓ Proved