Derivative of \( \displaystyle 5 \ln{\left(- \cot{\left(x \right)} + \csc{\left(x \right)} \right)} \)

Problem 2.1592 · hard Beautiful

1. \[ \frac{d}{d x} 5 \ln{\left(- \cot{\left(x \right)} + \csc{\left(x \right)} \right)} \]
   Start with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} \ln{\left(- \cot{\left(x \right)} + \csc{\left(x \right)} \right)} \]
   constant-multiplePull the constant out of the derivative.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} \left(- \cot{\left(x \right)} + \csc{\left(x \right)}\right)}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   chainApply the chain rule for the logarithm.✓ Proved
4. \[ = \frac{5 \left(- \frac{d}{d x} \cot{\left(x \right)} + \frac{d}{d x} \csc{\left(x \right)}\right)}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   sumDifferentiate the terms inside the parentheses separately.✓ Proved
5. \[ = \frac{5 \left(- \cot{\left(x \right)} \csc{\left(x \right)} + \csc^{2}{\left(x \right)}\right)}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   trig algebraApply the derivatives of cot(x) and csc(x). Simplify the signs inside the parentheses.✓ Proved
6. \[ = \frac{- 5 \cot{\left(x \right)} \csc{\left(x \right)} + 5 \csc^{2}{\left(x \right)}}{- \cot{\left(x \right)} + \csc{\left(x \right)}} \]
   algebraCombine the terms into a single fraction.✓ Proved
7. \[ = 5 \csc{\left(x \right)} \]
   algebra simplifyFactor out csc(x) from the numerator. Cancel the common term in the numerator and denominator.✓ Proved