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The given function is:

\[
y = \frac{a \sin x + b \cos x}{a \sin x - b \cos x}
\]

To find the derivative \( \frac{dy}{dx} \), we can apply the **quotient rule** for derivatives, which states:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}
\]

Here, \( u(x) = a \sin x + b \cos x \) and \( v(x) = a \sin x - b \cos x \).

### Step 1: Compute \( u'(x) \) and \( v'(x) \)

- \( u'(x) = a \cos x - b \sin x \)
- \( v'(x) = a \cos x + b \sin x \)

### Step 2: Apply the Quotient Rule

Substituting into the quotient rule formula:

\[
\frac{dy}{dx} = \frac{(a \sin x - b \cos x)(a \cos x - b \sin x) - (a \sin x + b \cos x)(a \cos x + b \sin x)}{(a \sin x - b \cos x)^2}
\]

### Step 3: Simplify the Numerator

Expanding both terms in the numerator:

1. \((a \sin x - b \cos x)(a \cos x - b \sin x)\)

\[
= a^2 \sin x \cos x - ab \sin^2 x - ab \cos^2 x + b^2 \cos x \sin x
\]
\[
= (a^2 + b^2) \sin x \cos x - ab (\sin^2 x + \cos^2 x)
\]
Since \( \sin^2 x + \cos^2 x = 1 \), this simplifies to:
\[
= (a^2 + b^2) \sin x \cos x - ab
\]

2. \((a \sin x + b \cos x)(a \cos x + b \sin x)\)

\[
= a^2 \sin x \cos x + ab \sin^2 x + ab \cos^2 x + b^2 \cos x \sin x
\]
\[
= (a^2 + b^2) \sin x \cos x + ab
\]

### Step 4: Combine the Two Terms in the Numerator

\[
\text{Numerator} = [(a^2 + b^2) \sin x \cos x - ab] - [(a^2 + b^2) \sin x \cos x + ab]
\]
\[
= -2ab
\]

### Step 5: Final Derivative

Now, we can write the derivative as:

\[
\frac{dy}{dx} = \frac{-2ab}{(a \sin x - b \cos x)^2}
\]

This is the required derivative of the given function.

Let me know if you would like more details on any of the steps! Here are some related questions:

1. How do you apply the quotient rule to a general function?
2. What is the physical interpretation of the derivative for trigonometric functions?
3. How do we differentiate products of trigonometric functions?
4. Can we use the chain rule to derive the result in another way?
5. What happens if \( a \) or \( b \) becomes zero in this context?

**Tip**: The quotient rule can simplify complex rational functions but always requires careful application of derivatives for both the numerator and the denominator.

Find the derivative of the following function, where a and b are nonzero constants: y = (a sin x + b cos x) / (a sin x - b cos x).

Learn how to find the derivative of the function y = (a sin x + b cos x) / (a sin x - b cos x) using the quotient rule. This problem covers key calculus concepts, including trigonometric differentiation and the quotient rule, suitable for Grade 12 or college-level students.

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