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

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

• A. (2x sin x - x^2 cos x)/sin^2(x)
• B. 2x cos(x)/sin^2(x)
• C. x^2/sin(x)^2
• D. 2/sin(x)

Answer: (A)

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

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