Differentiate the function f(x) = x^2/sin(x). What is the derivative?

To find the derivative of the function ( f(x) = \frac{x^2}{\sin(x)} ), we can use the quotient rule, which states that if you have a function in the form of ( \frac{u}{v} ), the derivative ( f'(x) ) is given by:

[

f'(x) = \frac{u'v - uv'}{v^2}

]

In this case, let ( u = x^2 ) and ( v = \sin(x) ). Now we need to find the derivatives of ( u ) and ( v ):

• The derivative ( u' = 2x )

• The derivative ( v' = \cos(x) )

Substituting these values into the quotient rule formula, we get:

[

f'(x) = \frac{(2x)(\sin(x)) - (x^2)(\cos(x))}{\sin^2(x)}

]

This simplifies to:

[

f'(x) = \frac{2x \sin(x) - x^2 \cos(x)}{\sin^2(x)}

]

When comparing this result to the provided options