23–51. Calculating derivatives Find the derivative of the following functions.
y = tan x + cot x

Verified step by step guidance

1. Step 1: Identify the functions involved in the expression. The given function is \( y = \tan x + \cot x \). Here, \( \tan x \) and \( \cot x \) are trigonometric functions.
2. Step 2: Recall the derivatives of the basic trigonometric functions. The derivative of \( \tan x \) is \( \sec^2 x \), and the derivative of \( \cot x \) is \( -\csc^2 x \).
3. Step 3: Apply the sum rule for derivatives. The sum rule states that the derivative of a sum of functions is the sum of their derivatives. Therefore, \( \frac{d}{dx}(\tan x + \cot x) = \frac{d}{dx}(\tan x) + \frac{d}{dx}(\cot x) \).
4. Step 4: Substitute the derivatives of the individual functions into the expression. This gives \( \sec^2 x - \csc^2 x \).
5. Step 5: Simplify the expression if possible. In this case, the expression \( \sec^2 x - \csc^2 x \) is already in its simplest form.