Differentiate the following functions w.r.t. x:
`(sin 3x)/( x-6)`

Answer

To differentiate the function \( y = \frac{\sin(3x)}{x - 6} \) with respect to \( x \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

where \( u = \sin(3x) \) and \( v = x - 6 \).

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = \sin(3x) \)
- \( v = x - 6 \)

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \cos(3x) \cdot \frac{d(3x)}{dx} = \cos(3x) \cdot 3 = 3\cos(3x) \]

2. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d(x - 6)}{dx} = 1 \]

### Step 3: Apply the Quotient Rule
Now, we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{(x - 6)(3\cos(3x)) - (\sin(3x))(1)}{(x - 6)^2} \]

### Step 4: Simplify the Expression
Now, simplify the numerator:

\[ \frac{dy}{dx} = \frac{(x - 6) \cdot 3\cos(3x) - \sin(3x)}{(x - 6)^2} \]

### Final Answer
Thus, the derivative of the function is:

\[ \frac{dy}{dx} = \frac{3(x - 6)\cos(3x) - \sin(3x)}{(x - 6)^2} \]

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