Step 1: Understanding the Concept:
To differentiate this function, it is easier to simplify the fraction by dividing each term in the numerator by the denominator before applying differentiation rules. Step 2: Key Formula or Approach:
1. Simplify: \(y = a \cot x + b + c \csc x\).
2. Derivatives: \(\frac{d}{dx}(\cot x) = -\csc^2 x\) and \(\frac{d}{dx}(\csc x) = -\csc x \cot x\). Step 3: Detailed Explanation:
Split the terms:
\[ y = \frac{a \cos x}{\sin x} + \frac{b \sin x}{\sin x} + \frac{c}{\sin x} \]
\[ y = a \cot x + b + c \csc x \]
Differentiate with respect to \(x\):
\[ \frac{dy}{dx} = a(-\csc^2 x) + 0 + c(-\csc x \cot x) \]
\[ \frac{dy}{dx} = -a \csc^2 x - c \csc x \cot x \] Step 4: Final Answer:
The derivative \( \frac{dy}{dx} \) is \( -a \csc^2 x - c \csc x \cot x \).