What is the simplified expression for the derivative of the function f(x) = x^2/sin(x)?

To find the derivative of the function f(x) = x^2/sin(x), you can apply the quotient rule, which states that if you have a function that is the ratio of two differentiable functions u(x) and v(x), the derivative f'(x) is given by:

f'(x) = (u'v - uv')/v^2.

For this function, let u = x^2 and v = sin(x). The derivatives of these functions are:

u' = 2x and v' = cos(x).

Now we can apply the quotient rule:

1. Calculate u'v:

u'v = (2x)(sin(x)) = 2x sin(x).

2. Calculate uv':

uv' = (x^2)(cos(x)).

3. Now substitute these into the quotient rule formula:

f'(x) = (2x sin(x) - x^2 cos(x)) / (sin^2(x)).

This expression captures the rate of change of f(x) with respect to x. The simplified form of the derivative, when organized, is indeed:

(2x sin(x) - x^2 cos(x)) / (sin^2