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

Problem 2.959 · hard Beautiful

1. \[ \frac{d}{d x} \frac{\ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} \]
   constant-multipleStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \ln{\left(\frac{1}{\cos^{2}{\left(2 x + 2 \right)}} \right)}}{4} \]
   chainMove the constant factor outside the derivative.✓ 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. \[ = \frac{\sin{\left(2 x + 2 \right)} \frac{d}{d x} \left(2 x + 2\right)}{2 \cos{\left(2 x + 2 \right)}} \]
   chainApply 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 simplifyDifferentiate the innermost linear function. Simplify the constant terms. Simplify the signs and constants. Combine the constant factor and simplify the powers of cosine. Combine the exponents of the cosine term.✓ Proved
7. \[ = \tan{\left(2 x + 2 \right)} \]
   simplifyUse the identity sin(u)/cos(u) = tan(u).✓ Proved