Find the derivative of the following functions.
y = cos x/sin x + 1

Verified step by step guidance

1. Step 1: Recognize that the function y = \frac{\cos x}{\sin x} + 1 can be rewritten as y = \cot x + 1, where \cot x is the cotangent function.
2. Step 2: Recall the derivative of the cotangent function: \frac{d}{dx}(\cot x) = -\csc^2 x.
3. Step 3: Differentiate the function y = \cot x + 1 with respect to x. The derivative of a constant (1 in this case) is 0.
4. Step 4: Apply the derivative rule from Step 2 to find the derivative of \cot x, which is -\csc^2 x.
5. Step 5: Combine the results from Steps 3 and 4 to express the derivative of the entire function: \frac{dy}{dx} = -\csc^2 x.