Final answer:
To differentiate the function f(x) = ln(100 sin²(x)), apply the chain rule and properties of logarithms to find f'(x) = 2 cot(x).
Step-by-step explanation:
To differentiate the given function f(x) = ln(100 sin²(x)), we first simplify it by understanding that natural logarithms and exponential functions are inverse functions. Thus, for example, In(ex) = x and eln(x) = x. This property helps us manipulate and differentiate functions involving natural logarithms more easily.
Applying the chain rule and the properties of logarithms, the derivative of f(x) with respect to x is found by differentiating the outside function (natural logarithm) and then multiplying by the derivative of the inside function (100 sin^2(x)). Recognizing that the logarithm of a product is the sum of the logarithms, and that the derivative of ln(u) is 1/u times the derivative of u, we have:
f'(x) = d/dx [ln(100) + ln(sin²(x))]
=f'(x) = 0 + d/dx [2 ln(sin(x))]
=f'(x) = 2*(1/sin(x))*(cos(x))
=f'(x) = 2 cot(x)