Mathematics

The key is recognizing that you're differentiating a composite function, so the chain rule is your main tool. Start by taking the derivative of the outer function (the natural log), which gives you $\frac{1}{\cot x - \csc x}$, then multiply by the derivative of the inner argument. The real insight is simplifying the derivatives of $\cot x$ and $\csc x$—once you compute those and combine them, the expression often collapses into something much cleaner than it first appears.

The key is recognizing that you're differentiating a composite function, so the chain rule is your main tool. Start by taking the derivative of the outer function (the natural log), which gives you $\frac{1}{\cot x - \csc x}$, then multiply by the derivative of the inner argument. The real insight is simplifying the derivatives of $\cot x$ and $\csc x$—once you compute those and combine them, the expression often collapses into something much cleaner than it first appears.