`cosec^(-1)3x`

Answer

To find the derivative of the function \( y = \csc^{-1}(3x) \), we can follow these steps:

### Step 1: Rewrite the function
Let \( y = \csc^{-1}(3x) \).

### Step 2: Use the derivative formula
We know that the derivative of \( \csc^{-1}(x) \) is given by:
\[ \frac{d}{dx} \csc^{-1}(x) = -\frac{1}{|x| \sqrt{x^2 - 1}} \]
In our case, we have \( x = 3x \).

### Step 3: Apply the chain rule
Using the chain rule, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx} \csc^{-1}(3x) = -\frac{1}{|3x| \sqrt{(3x)^2 - 1}} \cdot \frac{d}{dx}(3x) \]

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

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

### Step 6: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -\frac{3}{|3x| \sqrt{9x^2 - 1}} \]

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

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