find the derivative of the function f(y) =4y cos y + cot y

To find the derivative of the function f(y) = 4y \cos y + \cot y , we will differentiate each term separately using the following rules:

1. Derivative of \cos y = -\sin y
2. Derivative of \cot y = -\csc^2 y

Now, let's find the derivative:

f'(y) = \frac{d}{dy} (4y \cos y) + \frac{d}{dy} (\cot y)

f'(y) = 4 \cos y + 4y(-\sin y) + (-\csc^2 y)

f'(y) = 4 \cos y - 4y \sin y - \csc^2 y

Therefore, the derivative of the function f(y) = 4y \cos y + \cot y is f'(y) = 4 \cos y - 4y \sin y - \csc^2 y .

\boxed{f'(y) = 4 \cos y - 4y \sin y - \csc^2 y}