Final answer:
The function y = ln(sin² x / x) is differentiated using logarithmic properties and trigonometric identities. The result after simplification is y' = 2cot x - 1/x. This process involves the use of the chain rule and simplification of trigonometric expressions.
Step-by-step explanation:
Factoring and Differentiating Ln(sin² x / x)
To differentiate the function y = ln(sin² x / x) we can apply the properties of logarithms and trigonometric identities. The logarithm of a product, ln(xy) = ln x + ln y, and the logarithm of a quotient, ln(x/y) = ln x - ln y, are useful here. Applying these properties, we rewrite the function as y = ln(sin² x) - ln(x). Using the identity ln(Aˆ) = xlnA, the term ln(sin² x) simplifies to 2ln(sin x). Now we differentiate term by term.
The derivative of 2ln(sin x) using the chain rule is 2 * (1/sin x) * cos x, which simplifies to 2cot x. The derivative of -ln(x) is -1/x. So, the simplified derivative of the initial function is y' = 2cot x - 1/x.
Remember, it's necessary to be mindful of the original function's domain, since sin x must be non-zero, and x as well must be non-zero.