Derivative of \( \displaystyle - \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} \)

Problem 2.122 · hard

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} \]
   constant-multiplePull out the constant factor.✓ Proved
3. \[ = - \frac{\cos^{2}{\left(2 x + 2 \right)} \frac{d}{d x} \frac{1}{\cos^{2}{\left(2 x + 2 \right)}}}{4} \]
   logarithmic algebraApply the chain rule for the natural logarithm. Simplify the reciprocal of the negative power.✓ Proved
4. \[ = \frac{\frac{d}{d x} \cos{\left(2 x + 2 \right)}}{2 \cos{\left(2 x + 2 \right)}} \]
   powerApply the power rule to the inner function.✓ Proved
5. \[ = - \frac{\sin{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{2 \cos{\left(2 x + 2 \right)}} \]
   trigApply the chain rule to the cosine function.✓ Proved
6. \[ = - \frac{\sin{\left(2 x + 2 \right)}}{\cos{\left(2 x + 2 \right)}} \]
   derivative algebra algebra simplify algebraDifferentiate the innermost linear function. Simplify the signs and constants inside the expression. Combine the constant and the powers of cosine. Simplify the final expression. Rewrite the negative exponent as a fraction.✓ Proved
7. \[ = - \tan{\left(2 x + 2 \right)} \]
   trigUse the definition of the tangent function.✓ Proved