Find the derivative of f(x)= ln(|sin(x)|). Given that f(x) has a value for all x, state why the modulus is required.

The derivative can be found by using the chain rule. i.e. let g(x) = |sin(x)|, so f(x)=ln(g(x)), hence df/dx = df/dg * dg/dx

df/dg = 1/g, dg/dx = |cos(x)| so df/dx = |cos(x)|/|sin(x)|

For the second part, it is key to recognise that if y is negative then ln(y) is indeterminate. Hence if no modulus is present f(x) is indeterminate when sin(x) is negative.