`cosec^(-1)3x`

Answer

To differentiate the function \( y = \csc^{-1}(3x) \) with respect to \( x \), we will follow these steps:

### Step 1: Set up the equation
Let \( y = \csc^{-1}(3x) \).

### Step 2: Use the derivative formula for \( \csc^{-1}(y) \)
The derivative of \( \csc^{-1}(y) \) with respect to \( y \) is given by:
\[ \frac{dy}{dy} = -\frac{1}{|y| \sqrt{y^2 - 1}} \]
In our case, \( y = 3x \).

### Step 3: Differentiate with respect to \( x \)
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{dy} \cdot \frac{dy}{dx} \]
Substituting the derivative formula:
\[ \frac{dy}{dx} = -\frac{1}{|3x| \sqrt{(3x)^2 - 1}} \cdot \frac{d(3x)}{dx} \]

### Step 4: Calculate \( \frac{d(3x)}{dx} \)
The derivative of \( 3x \) with respect to \( x \) is:
\[ \frac{d(3x)}{dx} = 3 \]

### Step 5: Substitute back into the derivative
Now substituting this back into our expression:
\[ \frac{dy}{dx} = -\frac{1}{|3x| \sqrt{9x^2 - 1}} \cdot 3 \]

### Step 6: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -\frac{3}{|3x| \sqrt{9x^2 - 1}} \]
Since \( |3x| = 3|x| \), we can further simplify:
\[ \frac{dy}{dx} = -\frac{1}{|x| \sqrt{9x^2 - 1}} \]

### Final Result
Thus, the derivative of \( y = \csc^{-1}(3x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{1}{|x| \sqrt{9x^2 - 1}} \]

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