Derivative of ln(sin(x))

Differentiating ln(sin(x)) exercises 3 distinct techniques — the chain rule, the natural log rule and the trigonometric derivatives. Problems like this one are useful practice precisely because the rules have to be combined in the right order rather than applied in isolation.

The chain rule handles the composite structure here: differentiate the outer function while leaving the inner expression alone, then multiply by the derivative of that inner expression. Forgetting that final multiplication is one of the most common mistakes in differentiation. The natural logarithm differentiates to the reciprocal: d/dx[ln x] = 1/x. When the argument of the log is itself a function, the chain rule divides its derivative by that argument. The trigonometric derivatives follow the standard cycle: sine differentiates to cosine, cosine to negative sine, and tangent to secant squared. The sign on the cosine derivative is a frequent source of errors, so it is worth double-checking.

A good way to verify a derivative by hand is to compare it against the step-by-step breakdown above — each step names the rule being applied, so you can pinpoint exactly where your own work diverges if the answers differ.